I. D. Chueshov
Title:
Introduction to the Theory
of InfiniteDimensional
Dissipative Systems
«A CTA » 200
Author:
ISBN:
966–7021
966
7021–64
64–5
I. D. Chueshov
I ntroduction to the Theory
of Infinite-Dimensional
D issipative
S ystems
Universitylecturesincontemporarymathematics
This book provides an exhaustive introduction to the scope
of main ideas and methods of the
theory of infinite-dimensional dissipative dynamical systems which
has been rapidly developing in recent years. In the examples
systems generated by nonlinear
partial differential equations
arising in the different problems
of modern mechanics of continua
are considered. The main goal
of the book is to help the reader
to master the basic strategies used
in the study of infinite-dimensional
dissipative systems and to qualify
him/her for an independent scientific research in the given branch.
Experts in nonlinear dynamics will
find many fundamental facts in the
convenient and practical form
in this book.
The core of the book is composed of the courses given by the
author at the Department
of Mechanics and Mathematics
at Kharkov University during
a number of years. This book contains a large number of exercises
which make the main text more
complete. It is sufficient to know
the fundamentals of functional
analysis and ordinary differential
equations to read the book.
Translated by
You can O R D E R this book
while visiting the website
of «ACTA» Scientific Publishing House
http://www.acta.com.ua
www.acta.com.ua/en/
Constantin I. Chueshov
from the Russian edition («ACTA», 1999)
Translation edited by
Maryna B. Khorolska
Chapter
6
Homoclinic Chaos
in Infinite-Dimensional Systems
Contents
....§1
Bernoulli Shift as a Model of Chaos . . . . . . . . . . . . . . . . . . . . 365
....§2
Exponential Dichotomy and Difference Equations . . . . . . . 369
....§3
Hyperbolicity of Invariant Sets
for Differentiable Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 377
....§4
Anosov’s Lemma on e -trajectories . . . . . . . . . . . . . . . . . . . . 381
....§5
Birkhoff-Smale Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
....§6
Possibility of Chaos in the Problem
of Nonlinear Oscillations of a Plate . . . . . . . . . . . . . . . . . . . . 396
....§7
On the Existence of Transversal Homoclinic Trajectories . . 402
....
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
In this chapter we consider some questions on the asymptotic behaviour of a discrete dynamical system. We remind (see Chapter 1) that a discrete dynamical system is defined as a pair ( X , S ) consisting of a metric space X and a continuous
mapping of X into itself. Most assertions on the existence and properties of attractors given in Chapter 1 remain true for these systems. It should be noted that the following examples of discrete dynamical systems are the most interesting from the
point of view of applications: a) systems generated by monodromy operators (period
mappings) of evolutionary equations, with coefficients being periodic in time;
b) systems generated by difference schemes of the type t -1 ( u n + 1 - un ) =
= F ( un ) , n = 0 , 1 , 2 , ¼ in a Banach space X (see Examples 1.5 and 1.6 of Chapter 1).
The main goal of this chapter is to give a strict mathematical description of one
of the mechanisms of a complicated (irregular, chaotic) behaviour of trajectories.
We deal with the phenomenon of the so-called homoclinic chaos. This phenomenon
is well-known and is described by the famous Smale theorem (see, e.g., [1–3]) for finite-dimensional systems. This theorem is of general nature and can be proved for
infinite-dimensional systems. Its proof given in Section 5 is based on an infinite-dimensional variant of Anosov’s lemma on e -trajectories (see Section 4). The considerations of this Chapter are based on the paper [4] devoted to the finite-dimensional
case as well as on the results concerning exponential dichotomies of infinite-dimensional systems given in Chapter 7 of the book [5]. We follow the arguments given in [6]
while proving Anosov’s lemma.
§1
Bernoulli Shift as a Model of Chaos
Mathematical simulation of complicated dynamical processes which take place in real
systems requires that the notion of a state of chaos be formalized. One of the possible
approaches to the introduction of this notion relies on a selection of a class of explicitly solvable models with complicated (in some sense) behaviour of trajectories.
Then we can associate every model of the class with a definite type of chaotic behaviour and use these models as standard ones comparing their dynamical structure
with a qualitative behaviour of the dynamical system considered. A discrete dynamical system known as the Bernoulli shift is one of these explicitly solvable models.
Let m ³ 2 and let
ì
ü
S m = í x = ( ¼ , x-1 , x0 , x1 , ¼ ) : xj Î { 1 , 2 , ¼ , m } , j Î Z ý ,
î
þ
i.e. S m is a set of two-sided infinite sequences the elements of which are the integers 1 , 2 , ¼ , m . Let us equip the set S m with a metric
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Homoclinic Chaos in Infinite-Dimensional Systems
¥
d(x, y) =
å
i = -¥
xi - yi
2 - i ----------------------------- .
1 + xi - yi
(1.1)
Here x = { xi : i Î Z } and y = { yi : i Î Z } are elements of S m . Other methods
of introduction of a metric in Sm are given in Example 1.1.7 and Exercise 1.1.5.
E x e r c i s e 1.1 Show that the function d ( x , y ) satisfies all the axioms of
a metric.
E x e r c i s e 1.2 Let x = { x i } and y = { yi } be elements of the set S m .
Assume that xi = y i for i £ N and for some integer N . Prove that
d ( x , y ) £ 2- N + 1 .
E x e r c i s e 1.3 Assume that equation d ( x , y ) < 2 -N holds for x , y Î S m ,
where N is a natural number. Show that xi = yi for all i £ N - 1
(Hint: d ( x , y ) ³ 2 - i - 1 if x i ¹ yi ).
E x e r c i s e 1.4
Let x Î S m and let
U N ( x ) = { y Î S m : xi = yi for i £ N } .
(1.2)
Prove that for any 0 < e < 1 the relation
U N ( e ) + 2 ( x ) Ì { y : d ( x , y ) < e } Ì U N ( e ) - 1( x )
holds, where N(e) is an integer with the property
ln 1¤ e
N ( e ) < -------------- £ N ( e ) + 1 .
ln 2
E x e r c i s e 1.5 Show that the space Sm with metric (1.1) is a compact metric space.
In the space S m we define a mapping S which shifts every sequence one symbol
left, i.e.
[ S x ]i = xi + 1 ,
i ÎZ,
x = { xi } Î Sm .
Evidently, S is invertible and the relations
d ( S -1 x , S -1 y ) £ 2 d ( x , y )
d (S x, S y) £ 2 d (x, y) ,
hold for all x , y Î S m . Therefore, the mapping S is a homeomorphism.
The discrete dynamical system ( S m , S ) is called the Bernoulli shift of the
space of sequences of m symbols. Let us study the dynamical properties of the system ( S m , S ) .
E x e r c i s e 1.6 Prove that ( Sm , S ) has m fixed points exactly. What structure do they have?
Bernoulli Shift as a Model of Chaos
We call an arbitrary ordered collection a = ( a 1 , ¼ , aN ) with aj Î { 1 , ¼ , m }
a segment (of the length N ). Each element x Î Sm can be considered as an ordered infinite family of finite segments while the elements of the set S m can be constructed from segments. In particular, using the segment a = ( a1 , ¼ , aN ) we can
construct a periodic element a Î S m by the formula
aN k + j = aj ,
j Î {1 , ¼ , N} ,
k ÎZ.
(1.3)
Let a = ( a1 , ¼ , aN ) be a segment of the length N and let
a Î S m be an element defined by (1.3). Prove that a is a periodic
point of the period N of the dynamical system ( S m , S ) , i.e.
SN a = a .
E x e r c i s e 1.7
E x e r c i s e 1.8 Prove that for any natural N there exists a periodic point
of the minimal period equal to N .
E x e r c i s e 1.9 Prove that the set of all periodic points is dense in S m , i.e. for
every x Î Sm and e > 0 there exists a periodic point a with the
property d ( x , a ) < e (Hint: use the result of Exercise 1.4).
E x e r c i s e 1.10 Prove that the set of nonperiodic points is not countable.
E x e r c i s e 1.11 Let a = ( ¼ , a , a , a , ¼ ) and b = ( ¼ , b , b , b , ¼ ) be fixed points of the system ( S m , S ) . Let C = { ci } be an element
of Sm such that ci = a for i £ -M1 and c i = b for i ³ M2 , where
M1 and M2 are natural numbers. Prove that
lim S n c = a ,
n ® -¥
lim S n c = b .
n®¥
(1.4)
Assume that an element c Î S m possesses property (1.4) with c ¹ a and c ¹ b .
If a ¹ b , then the set
ga , b = { S n c : n Î Z }
is called a heteroclinic trajectory that connects the fixed points a and b .
If a = b , then ga = ga , a is called a homoclinic trajectory of the point a . The
elements of a heteroclinic (homoclinic, respectively) trajectory are called heteroclinic (homoclinic, respectively) points.
E x e r c i s e 1.12 Prove that for any pair of fixed points there exists an infinite
number of heteroclinic trajectories connecting them whereas the
corresponding set of heteroclinic points is dense in S m .
E x e r c i s e 1.13 Let
g1 = { S n a : n Î Z } = { S n a : n = 0 , 1 , ¼ , N1 - 1 }
and
g2 = { S n b : n Î Z } = { S n b : n = 0 , 1 , ¼ , N2 - 1 }
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Homoclinic Chaos in Infinite-Dimensional Systems
be cycles (periodic trajectories). Prove that there exists a heteroclinic trajectory g1 , 2 = { S n c : n Î Z } that connects the cycles g1
and g2 , i.e. such that
dist ( S n c , g1 ) º
inf d ( S n c , x ) ® 0 ,
x Î g1
n ® -¥
and
dist ( S n c , g2 ) = inf d ( S n c , x ) ® 0 ,
x Î g2
n ® +¥ .
For every N there exists only a finite number of segments of the length N . Therefore, the set L of all segments is countable, i.e. we can assume that L = { a k :
k = 1 , 2 , ¼ } , therewith the length of the segment a k + 1 is not less than the length
of ak . Let us construct an element b = { bi : i Î Z } from S m taking bi = 1 for
i £ 0 and sequentially putting all the segments ak to the right of the zeroth position. As a result, we obtain an element of the form
b = ( ¼ , 1 , 1 , 1 , a1 , a2 , a3 , ¼ ) ,
aj Î L .
(1.5)
E x e r c i s e 1.14 Prove that a positive semitrajectory g + = { S n b , n ³ 0 } with
b having the form (1.5) is dense in Sm , i.e. for every x Î S m and
e > 0 there exists n = n ( x , e ) such that d ( x , S n b ) < e .
E x e r c i s e 1.15 Prove that the semitrajectory g + constructed in Exercise 1.14
returns to an e -vicinity of every point x Î S m infinite number
of times (Hint: see Exercises 1.4 and 1.9).
E x e r c i s e 1.16 Construct a negative semitrajectory g – = { S n c : n £ 0 }
which is dense in S m .
Thus, summing up the results of the exercises given above, we obtain the following
assertion.
Theorem 1.1.
The dynamical system ( S m , S ) of the Bernoulli shift of sequences
of m symbols possesses the properties:
1) there exists a finite number of fixed points;
2) there exist periodic orbits of any minimal period and the set of these
orbits is dense in the phase space S m ;
3) the set of nonperiodic points is uncountable;
4) heteroclinic and homoclinic points are dense in the phase space;
5) there exist everywhere dense trajectories.
All these properties clearly imply the extraordinarity and complexity of the dynamics in the system ( S m , S ) . They also give a motivation for the following definitions.
Exponential Dichotomy and Difference Equations
Let ( X , f ) be a discrete dynamical system. The dynamics of the system ( X , f )
is called chaotic if there exists a natural number k such that the mapping f k is
topologically conjugate to the Bernoulli shift for some m , i.e. there exists a homeomorphism h: X ® Sm such that h ( f k ( x ) ) = S ( h ( x ) ) for all x Î X . We also say
that chaotic dynamics is observed in the system ( X , f ) if there exist a number k
and a set Y in X invariant with respect to f k ( f k Y Ì Y ) such that the restriction
of f k to Y is topologically conjugate to the Bernoulli shift.
It turns out that if a dynamical system ( X , f ) has a fixed point and a corresponding homoclinic trajectory, then chaotic dynamics can be observed in this system
under some additional conditions (this assertion is the core of the Smale theorem).
Therefore, we often speak about homoclinic chaos in this situation. It should also be
noted that the approach presented here is just one of the possible methods used to
describe chaotic behaviour (for example, other approaches can be found in [1] as
well as in book [7], the latter contains a survey of methods used to study the dynamics of complicated systems and processes).
§ 2 Exponential Dichotomy
and Difference Equations
This is an auxiliary section. Nonautonomous linear difference equations of the form
xn + 1 = An xn + h n ,
n ÎZ,
(2.1)
in a Banach space X are considered here. We assume that { An } is a family of linear
bounded operators in X , hn is a sequence of vectors from X . Some results both
on the dichotomy (splitting) of solutions to homogeneous ( hn º 0 ) equation (2.1)
and on the existence and properties of bounded solutions to nonhomogeneous equation are given here. We mostly follow the arguments given in book [5] as well as
in paper [4] devoted to the finite-dimensional case.
Thus, let { An : n Î Z } be a sequence of linear bounded operators in a Banach
space X . Let us consider a homogeneous difference equation
xn + 1 = An xn ,
n ÎJ,
(2.2)
where J is an interval in Z , i.e. a set of integers of the form
J = { n Î Z : m1 < n < m 2 } ,
where m1 and m 2 are given numbers, we allow the cases m1 = -¥ and m 2 = + ¥ .
Evidently, any solution { xn : n Î J } to difference equation (2.2) possesses the property
xm = F ( m , n ) xn ,
m ³ n,
m, n Î J ,
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Homoclinic Chaos in Infinite-Dimensional Systems
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where F ( m , n ) = A m - 1 × ¼ × An for m > n and F ( m , m ) = I . The mapping
F ( m , n ) is called an evolutionary operator of problem (2.2).
E x e r c i s e 2.1
Prove that for all m ³ n ³ k we have
F ( m , k) = F ( m , n) F ( n , k) .
6
E x e r c i s e 2.2 Let { Pn : n Î J } be a family of projectors (i.e. Pn2 = Pn ) in X
such that Pn + 1 A n = An Pn . Show that
Pm F ( m , n ) = F ( m , n ) Pn ,
m ³ n,
m, n Î J ,
i.e. the evolutionary operator F ( m , n ) maps Pn X into Pm X .
E x e r c i s e 2.3 Prove that solutions { x n } to nonhomogeneous difference
equation (2.1) possess the property
m-1
xm = F ( m , n ) xn +
å F(m, k + 1) h
k,
m > n.
k=n
Let us give the following definition. A family of linear bounded operators { An }
is said to possess an exponential dichotomy over an interval J with constants
K > 0 and 0 < q < 1 if there exists a family of projectors { Pn : n Î J } such that
a)
b)
Pn + 1 A n = A n Pn ,
F ( m , n ) Pn
£ K qm - n ,
n, n + 1 Î J ;
m ³ n,
m, n Î J ;
c) for n ³ m the evolutionary operator F ( n , m ) is a one-to-one mapping
of the subspace ( 1 - Pm ) X onto ( 1 - Pn ) X and the following estimate
holds:
F ( n , m ) -1 (1 - Pn )
£ K qn - m ,
m £ n,
m, n Î J .
If these conditions are fulfilled, then it is also said that difference equation (2.2) admits an exponential dichotomy over J . It should be noted that the cases J = Z and
J = Z ± are the most interesting for further considerations, where Z+ ( Z– ) is
the set of all nonnegative (nonpositive) integers.
The simplest case when difference equation (2.2) admits an exponential dichotomy is described in the following example.
E x a m p l e 2.1 (autonomous case)
Assume that equation (2.2) is autonomous, i.e. An º A for all n , and the spectrum s ( A ) does not intersect the unit circumference { z Î C : z =1 } . Linear
operators possessing this property are often called hyperbolic (with respect to
the fixed point x = 0 ). It is well-known (see, e.g., [8]) that in this case there
exists a projector P with the properties:
Exponential Dichotomy and Difference Equations
a) A P = P A , i.e. the subspaces PX and ( 1 - P ) X are invariant with respect to A ;
b) the spectrum s ( A PX ) of the restriction of the operator A to P X lies
strictly inside of the unit disc;
c) the spectrum s ( A ( 1 - P ) X ) of the restriction of A to the subspace
( 1 - P ) X lies outside the unit disc.
E x e r c i s e 2.4 Let C be a linear bounded operator in a Banach space X and
let r º max { z : z Î s ( C ) } be its spectral radius. Show that for
any q > r there exists a constant Mq ³ 1 such that
C n £ Mq q n ,
n = 0, 1, 2, ¼
(Hint: use the formula r = lim C n 1/ n the proof of which can be
n®¥
found in [9], for example).
Applying the result of Exercise 2.4 to the restriction of the operator A to PX , we obtain that there exist K > 0 and 0 < q < 1 such that
An P £ K q n ,
n ³ 0.
(2.3)
It is also evident that the restriction of the operator A to ( 1 - P ) X is invertible and
the spectrum of the inverse operator lies inside the unit disc. Therefore,
A-n (1 - P)
£ K qn ,
n ³ 0,
(2.4)
where the constants K > 0 and 0 < q < 1 can be chosen the same as in (2.3). The
evolutionary operator F ( m , n ) of the difference equation xn + 1 = A xn has the
form F ( m , n ) = A m - n , m ³ n . Therefore, the equality AP = PA and estimates
(2.3) and (2.4) imply that the equation xn + 1 = A xn admits an exponential dichotomy over Z , provided the spectrum of the operator A does not intersect the unit
circumference.
E x e r c i s e 2.5 Assume that for the operator A there exists a projector P
such that AP = PA and estimates (2.3) and (2.4) hold with 0 < q < 1 .
Show that the spectrum of the operator A does not intersect the
unit circumference, i.e. A is hyperbolic.
Thus, the hyperbolicity of the linear operator A is equivalent to the exponential dichotomy over Z of the difference equation xn + 1 = A xn with the projectors Pn independent of n . Therefore, the dichotomy property of difference equation (2.2)
should be considered as an extension of the notion of hyperbolicity to the nonautonomous case. The meaning of this notion is explained in the following two exercises.
E x e r c i s e 2.6 Let A be a hyperbolic operator. Show that the space X can be
decomposed into a direct sum of stable X s and unstable X u subspaces, i.e. X = X s + X u therewith
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Homoclinic Chaos in Infinite-Dimensional Systems
An x £ K q n x ,
An x ³ K -1 q -n x ,
x Î Xs ,
x Î Xu ,
n ³ 0,
n ³ 0,
with some constants K > 0 and 0 < q < 1 .
E x e r c i s e 2.7 Let X = R 2 be a plane and let A be an operator defined by
the formula
A ( x1 , x2 ) = ( 2 x1 + x2 ; x1 + x2 ) ,
x = ( x1 ; x2 ) Î R 2 .
Show that the operator A is hyperbolic. Evaluate and display graphically stable X s and unstable X u subspaces on the plane. Display
graphically the trajectory { An x : n Î Z } of some point x that lies
neither in X s , nor in X u .
The next assertion (its proof can be found in the book [5]) plays an important role in
the study of existence conditions of exponential dichotomy of a family of operators
{ An : n Î Z } .
Theorem 2.1.
Let { An : n Î Z } be a sequence of linear bounded operators in a Banach space X . Then the follo
following assertions are equivalent:
(i) the sequence { An : n Î Z } possesses an exponential dichotomy over
Z,
(ii) for any bounded sequence { hn : n Î Z } from X there exists a unique
bounded solution { xn : n Î Z } to the nonhomogeneous difference
equation
xn + 1 = An xn + hn ,
n ÎZ.
(2.5)
In the case when the sequence { A n } possesses an exponential dichotomy, solutions
to difference equation (2.5) can be constructed using the Green function :
ì F(n, m) P ,
m
ï
G (n, m) = í
ï -[ F ( m , n ) ] -1 ( 1 - P ) ,
m
î
E x e r c i s e 2.8
n ³ m,
n < m.
Prove that G ( n , m ) £ K q n - m .
E x e r c i s e 2.9 Prove that for any bounded sequence { hn : n Î Z } from X
a solution to equation (2.5) has the form
xn =
å G (n , m + 1) h
m ÎZ
m,
n ÎZ.
Exponential Dichotomy and Difference Equations
Moreover, the following estimate is valid:
1+ q
£ K ------------ sup hn .
1- q n
sup xn
n
The properties of the Green function enable us to prove the following assertion
on the uniqueness of the family of projectors { Pn } .
Lemma 2.1.
Let a sequence { An } possess an exponential dichotomy over Z . Then
the projectors { Pn : n Î Z } are uniquely defined.
Proof.
Assume that there exist two collections of projectors { Pn } and { Q n } for
which the sequence { A n } possesses an exponential dichotomy. Let GP ( n , m )
and GQ ( n , m ) be Green functions constructed with the help of these collections. Then Theorem 2.1 enables us to state (see Exercise 2.9) that
åG
P (n,
m + 1 ) hm =
m ÎZ
åG
Q (n,
m + 1 ) hm
m ÎZ
for all n Î Z and for any bounded sequence { hn } Ì X . Assuming that h m = 0
for m ¹ k - 1 and hm = h for m = k - 1 , we find that
GP ( n , k ) h = GQ ( n , k ) h ,
h ÎX,
n, k Î Z ,
n ³ k.
This equality with n = k gives us that Pn h = Qn h . Thus, the lemma is proved.
In particular, Theorem 2.1 implies that in order to prove the existence of an exponential dichotomy it is sufficient to make sure that equation (2.5) is uniquely solvable for any bounded right-hand side. It is convenient to consider this difference
¥
equation in the space lX º l ¥ ( Z , X ) of sequences x = { x n : n Î Z } of elements of X
for which the norm
x l ¥ = { xn }
l¥
ì
ü
= sup í xn : n Î Z ý
î
þ
(2.6)
is finite. Assume that the condition
ì
ü
sup í An : n Î Z ý < ¥
î
þ
(2.7)
¥
¥
is valid. Then for any x = { xn } Î lX the sequence { yn = xn - A n - 1 xn - 1 } lies in lX .
Consequently, equation
( L x ) n = xn - An -1 xn -1 ,
x = { xn } Î lX¥
¥
lX
l ¥(Z,
(2.8)
=
X ) . Therewith asdefines a linear bounded operator acting in the space
sertion (ii) of Theorem 2.1 is equivalent to the assertion on the invertibility of the
operator L given by equation (2.8).
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The assertion given below provides a sufficient condition of invertibility of the
operator L . Due to Theorem 2.1 this condition guarantees the existence of an exponential dichotomy for the corresponding difference equation. This assertion will be
used in Section 4 in the proof of Anosov’s lemma. It is a slightly weakened variant
of a lemma proved in [6].
6
Theorem 2.2.
Assume that a sequence of operators { A n : n Î Z } satisfies condition
(2.7).. Let there exist a family of projectors { Q n : n Î Z } such that
1 - Qn £ K ,
Qn £ K ,
(2.9)
Qn + 1 A n ( 1 - Qn ) £ d ,
( 1 - Qn + 1 ) An Qn £ d ,
(2.10)
for all n Î Z . We also assume that the operator ( 1 - Qn + 1 ) An is invertible
as a mapping from ( 1 - Q n ) X into ( 1 - Qn + 1 ) X and the estimates
[ ( 1 - Q n + 1 ) An ] -1 ( 1 - Q n + 1 ) £ l
An Qn £ l ,
(2.11)
are valid for every n Î Z . If
--- ,
--- ,
(2.12)
d £ 1
Kl £ 1
8
8
¥
then the operator L acting in lX according to formula (2.8) is invertible
and L -1 £ 2 K + 1 .
Proof.
Let us first prove the injectivity of the mapping L . Assume that there exists a
¥
nonzero element x = { xn} Î lX such that L x = 0 , i.e. xn = An - 1 x n - 1 for all n Î Z .
Let us prove that the sequence { xn } possesses the property
( 1 - Qn ) xn £ Qn xn
(2.13)
for all n Î Z . Indeed, let there exist m Î Z such that
( 1 - Qm ) xm > Qm xm .
(2.14)
It is evident that this equation is only possible when ( 1 - Q m ) x m > 0 . Let us consider the value
Nm + 1 º ( 1 - Qm + 1 ) xm + 1 - Qm + 1 x m + 1 =
= ( 1 - Qm + 1 ) Am xm - Qm + 1 Am xm .
(2.15)
It is clear that
( 1 - Qn + 1 ) An xn
³ ( 1 - Qn + 1 ) An ( 1 - Qn ) xn - ( 1 - Qn + 1 ) An Qn xn .
Since
[ ( 1 - Q n + 1 ) An ] -1 ( 1 - Q n + 1 ) An ( 1 - Qn ) = ( 1 - Qn ) ,
Exponential Dichotomy and Difference Equations
it follows from (2.11) that
( 1 - Qn ) x
£ l ( 1 - Q n + 1 ) An ( 1 - Q n ) x
for every x Î X and for all n Î Z . Therefore, we use estimates (2.10) to find that
( 1 - Q n + 1 ) An xn
³ l -1 ( 1 - Q n ) xn - d xn .
(2.16)
Then it is evident that
Qn + 1 An xn
£ Q n + 1 An Qn xn + Q n + 1 An ( 1 - Q n ) xn
£
£ æ Q n + 1 × An Q n + Qn + 1 An ( 1 - Qn ) ö xn .
è
ø
Therefore, estimates (2.9)–(2.11) imply that
Qn + 1 An xn
£ ( K l + d ) xn ,
n ÎZ.
(2.17)
Thus, equations (2.15)–(2.17) lead us to the estimate
N m + 1 ³ l -1 ( 1 - Q m ) x m - ( 2 d + K l ) xm .
It follows from (2.14) that
xm
£ Qm xm + ( 1 - Qm ) xm < 2 ( 1 - Q m ) xm .
Therefore,
N m + 1 > ( l -1 - 2 K l - 4 d ) ( 1 - Q m ) xm .
Hence, if conditions (2.12) hold, then
( 1 - Qm + 1 ) xm + 1 - Qm + 1 xm + 1 > 7 ( 1 - Qm ) xm > 0 .
(2.18)
When proving (2.18) we use the fact that
l -1 ³ 8 K ³ 8 Qn ³ 8 .
Thus, equation (2.18) follows from (2.14), i.e. N m > 0 implies Nm + 1 > 0 . Hence,
( 1 - Qn ) xn > Qn xn
for all
n ³ m.
Moreover, (2.18) gives us that
K × xn ³ ( 1 - Q n ) xn ³ 7 n - m ( 1 - Qm ) xm ,
n ³ m.
Therefore, xn ® + ¥ as n ® + ¥ . This contradicts the assumption x = { xn } Î lX¥ .
Thus, for all n Î Z estimate (2.13) is valid. In particular it leads us to the inequality
xn
£ ( 1 - Qn ) xn + Qn xn £ 2 Qn xn .
(2.19)
Therefore, it follows from (2.17) that
Qn + 1 xn + 1 = Qn + 1 A n xn £ 2 ( K l + d ) Qn x n
for all n Î Z . We use conditions (2.12) to find that
--- Q n xn ,
Qn + 1 xn + 1 £ 1
2
n ÎZ.
(2.20)
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If x = { xn } ¹ 0 , then inequality (2.19) gives us that there exists m Î Z such that
Q m xm ¹ 0 . Therefore, it follows from (2.20) that
6
Qn xn ³ 2 m - n Qm xm > 0
for all n £ m . We tend n ® -¥ to obtain that Q n xn ® + ¥ which is impossible
due to (2.9) and the boundedness of the sequence { xn } . Therefore, there does not
¥
exist a nonzero x Î lX such that L x = 0 . Thus, the mapping L is injective.
Let us now prove the surjectivity of L . Let us consider an operator R in the
¥
space lX acting according to the formula
y = { yn } Î lX¥ ,
( R y) n = Qn yn - Bn ( 1 - Q n + 1 ) yn + 1 ,
where the operator Bn = [ ( 1 - Q n + 1 ) An ] -1 acts from ( 1 - Q n + 1 ) X into ( 1 - Qn ) X
and is inverse to ( 1 - Qn + 1 ) A n
. It follows from (2.9) and (2.11) that
( 1 - Qn ) X
Ry
l¥
£ (K + l) y
l¥
,
y Î lX¥ .
(2.21)
It is evident that
( L R y )n - yn = -( 1 - Q n ) yn - Bn ( 1 - Q n + 1 ) yn + 1 - An - 1 Qn - 1 yn - 1 + An - 1 Bn - 1 ( 1 - Q n ) y n .
Since
( 1 - Qn ) An - 1 Bn - 1(1 - Qn) = 1 - Qn ,
we have that
( L R y ) n - yn = - Bn ( 1 - Q n + 1 ) yn + 1 - An - 1 Qn - 1 yn - 1 +
+ Qn An - 1 ( 1 - Q n - 1 ) Bn - 1 ( 1 - Qn ) yn .
Consequently,
( L R y ) n - yn
£
Bn ( 1 - Q n + 1 ) × yn + 1 + An - 1 Q n - 1 × yn - 1 +
+ Q n A n - 1 ( 1 - Q n - 1 ) × Bn - 1 ( 1 - Q n ) × yn .
Therefore, inequalities (2.10), (2.11), and (2.12) give us that
1
L R y - y l ¥ £ l ( 2 + d ) y l ¥ £ --- y l ¥ ,
2
i.e. L R - 1 £ 1 ¤ 2 . That means that the operator L R is invertible and
( L R ) -1 £ ( 1 - L R - 1 ) -1 £ 2 .
(2.22)
lX¥ . Then it is evident that the element
Let h = { hn } be an arbitrary element of
y=
= R ( L R ) -1 h is a solution to equation L y = h . Moreover, it follows from (2.21) and
(2.22) that
y
l¥
£ 2 (K + l) h
l¥
.
Hence, L is surjective and L -1 £ 2 K + 1 . Theorem 2.2 is proved.
Hyperbolicity of Invariant Sets for Differentiable Mappings
§3
Hyperbolicity of Invariant Sets
for Differentiable Mappings
Let us remind the definition of the differentiable mapping. Let X and Y be Banach
spaces and let U be an open set in X . The mapping f from U into Y is called
(Frechét) differentiable at the point x Î U if there exists a linear bounded operator D f ( x ) from X into Y such that
lim
v ®0
1-----f (x + v) - f (x) - D f (x) v = 0 .
v
If the mapping f is differentiable at every point x Î U , then the mapping D f :
x ® D f ( x ) acts from U into the Banach space L ( X , Y ) of all linear bounded operators from X into Y . If D f : U ® L ( X , Y ) is continuous, then the mapping f is
said to be continuously differentiable (or C 1 -mapping) on U . The notion of
the derivative of any order can also be introduced by means of induction. For example,
D 2 f ( x ) is the Frechét derivative of the mapping D f : U ® L ( X , Y ) .
Let g and f be continuously differentiable mappings from
U Ì X into Y and from W Ì Y into Z , respectively. Moreover, let
U and W be open sets such that g( U ) Ì W . Prove that ( f ° g) ( x ) =
= f ( g ( x ) ) is a C 1 -mapping on U and obtain a chain rule for the
differentiation of a composed function
E x e r c i s e 3.1
D( f ° g) (x) = D f (g (x) ) D g (x) ,
x ÎU.
E x e r c i s e 3.2 Let f be a continuously differentiable mapping from X into
X and let f n be the n -th degree of the mapping f , i.e. f n ( x ) =
= f ( f n - 1 ( x ) ) , n ³ 1 , f 1 ( x ) º f ( x ) . Prove that f n is a C 1 -mapping on X and
( D f n ) ( x ) = D f ( f n - 1( x ) ) × D f ( f n - 2 ( x ) ) × ¼ × D f ( x ) . (3.1)
Now we give the definition of a hyperbolic set. Assume that f is a continuously differentiable mapping from a Banach space X into itself and L is a subset in X which
is invariant with respect to f ( f ( L ) Ì L ) . The set L is called hyperbolic (with
respect to f ) if there exists a collection of projectors { P ( x ) : x Î L } such that
a) P ( x ) continuously depends on x Î L with respect to the operator
norm;
b) for every x Î L
D f (x) × P (x) = P ( f (x ) ) × D f (x) ;
(3.2)
c) the mappings D f ( x ) are invertible for every x Î L as linear operators
from ( 1 - P ( x ) ) X into ( I - P ( f ( x ) ) ) X ;
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d) for every x Î L the following equations hold:
( D f n ) ( x ) P ( x ) £ K qn ,
[ (D f n) (x) ]
-1
n ³ 0,
( 1 - P ( f n ( x ) ) ) £ K qn ,
(3.3)
n ³ 0,
(3.4)
with the constants K > 0 and 0 < q < 1 independent of x Î L . Here
f n is the n -th degree of the mapping f ( f n ( x ) = f ( f n - 1 ( x ) ) for
n ³ 1 and f 1 ( x ) º f ( x ) ).
It should be noted that properties (b) and (c) as well as formula (3.1) enable us
to state that ( D f n ) ( x ) maps ( 1 - P ( x ) ) X into ( 1 - P ( f n ( x ) ) ) and is an invertible
operator. Therefore, the value in the left-hand side of inequality (3.4) exists.
E x e r c i s e 3.3 Let L = { x 0 } , where x 0 is a fixed point of a C 1 -mapping f ,
i.e. f ( x 0 ) = x0 . Then for the set L to be hyperbolic it is necessary
and sufficient that the spectrum of the linear operator D f ( x0 ) does
not intersect the unit circumference (Hint: see Example 2.1).
Let L be an invariant hyperbolic set of a C 1 -mapping f and let g = { xn : n Î Z }
be a complete trajectory (in L ) for f , i.e. g = { xn } is a sequence of points from L
such that f ( x n ) = xn + 1 for all n Î Z . Let us consider a difference equation obtained as a result of linearization of the mapping f along g :
un + 1 = D f ( xn ) un ,
n ÎZ.
(3.5)
E x e r c i s e 3.4 Prove that the evolutionary operator F ( m , n ) of difference
equation (3.5) has the form
F ( m , n ) = ( D f m - n ) ( xn ) ,
m > n,
m, n Î Z .
E x e r c i s e 3.5 Prove that difference equation (3.5) admits an exponential dichotomy over Z with (i) the constants K and q given by equations
(3.3) and (3.4) and (ii) the projectors Pn = P ( xn ) involved in the
definition of the hyperbolicity.
It should be noted that property (a) of uniform continuity implies that the projectors
P ( x ) are similar to one another, provided the values of x are close enough.
The proof of this fact is based on the following assertion.
Lemma 3.1.
Let P and Q be projectors in a Banach space X . Assume that
1P - Q < -----2K
for some constant K ³ 1 . Then the operator
P £ K,
1- P £ K ,
J = P Q + ( 1 - P ) ( 1 - Q)
(3.6)
(3.7)
Hyperbolicity of Invariant Sets for Differentiable Mappings
possesses the property P J = J Q and is invertible, therewith
J -1 £ ( 1 - 2 K × P - Q ) -1 .
(3.8)
Proof.
Since P 2 + ( 1 - P ) 2 = 1 , we have
J - 1 = J - P 2 - (1 - P)2 = P (Q - P) + (1 - P) (P - Q) .
It follows from (3.6) that
J -1 £ 2K P-Q < 1 .
Hence, the operator J -1 can be defined as the following absolutely convergent
series
J -1 =
¥
å (1 - J)
n
.
n=0
This implies estimate (3.8). The permutability property PJ = J Q is evident.
Lemma 3.1 is proved.
E x e r c i s e 3.6 Let L be a connected compact set and let { P ( x ) : x Î L } be
a family of projectors for which condition (a) of the hyperbolicity definition holds. Then all operators P ( x ) are similar to one another, i.e.
for any x , y Î L there exists an invertible operator J = Jx , y such
that P ( x ) = J P ( y ) J -1 .
The following assertion contains a description of a situation when the hyperbolicity
of the invariant set is equivalent to the existence of an exponential dichotomy for difference equation (3.5) (cf. Exercise 3.5).
Theorem 3.1.
Let f ( x ) be a continuously differentiable mapping of the space X into
itself. Let x0 be a hyperbolic fixed point of f ( f ( x 0 ) = x0 ) and let { yn : n Î Z}
be a homoclinic trajectory (not equal to x0 ) of the mapping f , i.e.
f ( yn ) = yn + 1 ,
n ÎZ,
yn ® x0 ,
n ® ±¥ .
(3.9)
Then the set L = { x0 } È { yn : n Î Z } is hyperbolic if and only if the difference equation
un + 1 = D f ( yn ) un ,
n ÎZ,
(3.10)
possesses an exponential dichotomy over Z .
Proof.
If L is hyperbolic, then (see Exercise 3.5) equation (3.10) possesses an exponential dichotomy over Z . Let us prove the converse assertion. Assume that equation (3.10) possesses an exponential dichotomy over Z with projectors { Pn : n Î Z }
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and constants K and q . Let us denote the spectral projector of the operator D f ( x0 )
corresponding to the part of the spectrum inside the unit disc by P . Without loss
of generality we can assume that
[ D f ( x0 ) ] n P £ K qn ,
6
n ³ 0,
[ D f ( x0 ) ] -n ( 1 - P ) £ K q -n ,
n ³ 0.
Thus, for every x Î L the projector P ( x ) is defined: P ( x0 ) = P , P ( yk ) = Pk .
The structure of the evolutionary operator of difference equation (3.10) (see Exercise 3.4) enables us to verify properties (b)–(d) of the definition of a hyperbolic set.
In order to prove property (a) it is sufficient to verify that
Pk - P ® 0
k ® ±¥ .
as
(3.11)
Since L is a compact set, then
M = sup { D f ( x ) : x Î L } < ¥ .
(3.12)
Let us consider the following difference equations
vn + 1 = D f ( x0 ) vn ,
n ÎZ,
(3.13)
and
wn( k+) 1 = D f ( yn + k ) wn( k ) ,
n ÎZ,
(3.14)
where k is an integer. It is evident that equation (3.14) admits an exponential dichotomy over Z with constants K and q and projectors Pn( k ) = Pn + k . Let G ( n , m )
and G ( k ) ( n , m ) be the Green functions (see Section 2) of difference equations
(3.13) and (3.14). We consider the sequence
xn = G ( n , 0 ) z - G ( k ) ( n , 0 ) z ,
z ÎX.
Since (see Exercise 2.8)
G(n, 0) £ K q n ,
G(k) (n , 0) £ K q n ,
(3.15)
we have that the sequence { xn } is bounded. Moreover, it is easy to prove (see Exercise 2.9) that { xn } is a solution to the difference equation
xn + 1 - D f ( x0 ) xn = hn º [ D f ( x0 ) - D f ( yn + k ) ] G ( k ) ( n , 0 ) z .
It follows from (3.12) and (3.15) that the sequence { hn } is bounded. Therefore,
(see Exercise 2.9),
G ( n , 0 ) z - G ( k ) ( n , 0 ) z º xn =
å G (n, m + 1) h
m.
m ÎZ
If we take n = 0 in this formula, then from the definition of the Green function we
obtain that
( P - Pk ) z =
å G(0, m + 1) (D f (x ) - D f (y
0
m ÎZ
m + k))
G( k) ( m , 0 ) z .
H y p e r b o l i c i t y o f I n v a r i a n tA nS oe st o
s vf’osr LDe im
f fm
e rae on nt i aeb-lter aMj ea cptpoi rnigess
Therefore, equation (3.15) implies that
( P - Pk ) z
£ K2
å
D f ( x0 ) - D f ( ym + k ) × q
m + m +1
× z .
m ÎZ
Consequently,
P - Pk
£ K2
å
D f ( x0 ) - D f ( ym + k ) × q2 m - 1 £
m ÎZ
ì
£ K 2 í max D f ( x0 ) - D f ( ym + k ) ×
îm £ N
å
q2 m -1 + 2 M
m £ N
ü
q 2 m -1 ý ,
þ
m >N
å
where N is an arbitrary natural number. Upon simple calculations we find that
P - Pk
2 K 2 - æ max D f ( x ) - D f ( y
2 N + 2ö
£ ---------------------0
m + k) + 2 M q
ø
q (1 - q 2) è m £ N
for every N ³ 1 . It follows that
lim
k ® ±¥
P - Pk
4K2M
£ ----------------- q 2 N + 1 ,
1- q2
N = 1, 2, ¼ .
We assume that N ® + ¥ to obtain that
lim
k ® ±¥
P - Pk £ 0 .
This implies equation (3.11). Therefore, Theorem 3.1 is proved.
It should be noted that in the case when the set L = { x0 } È { yn : n Î Z } from Theorem 3.1 is hyperbolic the elements yn of the homoclinic trajectory g = { yn : n Î Z }
are called transversal homoclinic points . The point is that in some cases (see,
e.g., [4]) it can be proved that the hyperbolicity of L is equivalent to the transversality property at every point yn of the stable W s ( x0 ) and unstable W u ( x0 ) manifolds of a fixed point x0 (roughly speaking, transversality means that the surfaces
W s ( x0 ) and W u ( x0 ) intersect at the point yn at a nonzero angle). In this case the
trajectory g is often called a transversal homoclinic trajectory .
§4
Anosov’s Lemma on e -trajectories
Let f be a C 1 -mapping of a Banach space X into itself. A sequence { yn : n Î Z }
in X is called a d -pseudotrajectory (or d -pseudoorbit) of the mapping f if for
all n Î Z the equation
yn + 1 - f ( yn )
£ d
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is valid. A sequence { xn : n Î Z } is called an e -trajectory of the mapping f corresponding to a d -pseudotrajectory { yn : n Î Z } if
(a) f ( xn ) = xn + 1 for any n Î Z ;
6
(b) xn - yn £ e for all n Î Z .
It should be noted that condition (a) means that { xn } is an orbit (complete trajectory) of the mapping f . Moreover, if a pair of C 1 -mappings f and g is given,
then the notion of the e -trajectory of the mapping g corresponding to a d -pseudoorbit of the mapping f can be introduced.
The following assertion is the main result of this section.
Theorem 4.1.
Let f be a C 1-mapping of a Banach space X into itself and let L be
a hyperbolic invariant ( f ( L ) Ì L ) set. Assume that there exists a D -vicinity O of the set L such that f ( x ) and D f ( x ) are bounded and uniformly
continuous on the closure O of the set O . Then there exists e0 > 0 possessing the property that for every 0 < e £ e0 there exists d = d ( e ) > 0 such
that any d -pseudoorbit { yn : n Î Z } lying in L has a unique e -trajectory
{ xn : n Î Z } corresponding to { yn } .
As the following theorem shows, the property of the mapping f to have an e -trajectory is rough, i.e. this property also remains true for mappings that are close to f .
Theorem 4.2.
Assume that the hypotheses of Theorem 4.1 hold for the mapping f .
Let Wh ( f ) be a set of continuously differentiable mappings g of the space
X into itself such that the following estimates hold on the closure O of the
D -vicinity O of the set L :
f (x) - g (x) < h ,
D f (x) - D g (x) < h .
(4.1)
Then e0 > 0 can be chosen to possess the property that for every e Î ( 0 , e0 ]
there exist d = d ( e ) > 0 and h = h ( e ) > 0 such that for any d -pseudotrajectory { yn : n Î Z } (lying in L ) of the mapping f and for any g Î Wh ( f )
there exists a unique trajectory { xn : n Î Z } of the mapping g with the property
yn - x n £ e
for all
n ÎZ.
It is clear that Theorem 4.1 is a corollary of Theorem 4.2 the proof of which is based
on the lemmata below.
H y p e r b o l i c i t y o f I n v a r i a n tA nS oe st o
s vf’osr LDe im
f fm
e rae on nt i aeb-lter aMj ea cptpoi rnigess
Lemma 4.1.
Let U be an open set in a Banach space X and let F : U ® X be a continuously differentiable mapping. Assume that for some point y Î U
there exist an operator [ D F ( y ) ] -1 and a number e0 > 0 such that
D F (x) - D F (y)
£ æ 2 [ D F ( y ) ] -1 ö
è
ø
-1
(4.2)
for all x with the property x - y £ e 0 . Assume that for some e Î( 0 , e0 ]
the inequality
£ q × e æ2 [ D F (y) ]
è
F (y)
-1
ö
ø
-1
(4.3)
is valid with 0 < q < 1 . Then for any C 1 -mapping G : U ® X such that
£ e ( 1 - q ) æ 2 [ D F ( y ) ] -1 ö
è
ø
G (x) - F (x)
-1
(4.4)
and
--- æ 2 [ D F ( y ) ] -1 ö
£ 1
ø
2è
DG (x) - D F (x)
-1
(4.5)
for x - y £ e 0 , the equation G ( x ) = 0 has a unique solution x with
the property x - y £ e .
Proof.
-1 -1
Let G = ( 2 [ D F ( y ) ] ) and let
h (x) = F (x) - F (y) - D F (y) (x - y) .
For x1 , x2 Î Be º { z :
0
z - y £ e0 } we have that
h ( x1 ) - h ( x2 ) = F ( x1 ) - F ( x2 ) - D F ( y ) ( x1 - x2 ) =
1
=
ò ( D F ( x + x ( x - x ) ) - D F ( y ) ) ( x - x ) dx .
1
2
1
1
2
0
Since
1
h ( x1 ) - h ( x2 )
£
ò D F (x + x (x - x ) ) - D F (y) dx
1
2
1
x1 - x2 ,
0
it follows from (4.2) that
h ( x1 ) - h ( x 2 ) £ G x1 - x2
(4.6)
for all x1 and x2 from Be . Now we rewrite the equation G ( x ) = 0 in the form
0
x = T ( x ) º y - [ D F ( y ) ] -1 ( G ( x ) - F ( x ) + F ( y ) + h ( x ) ) .
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Let us show that the mapping T has a unique fixed point in the ball Be = { x :
x - y £ e } . It is evident that
T(x) - y
£
[ D F ( y ) ] -1 æ G ( x ) - F ( x ) + F ( y ) + h ( x ) ö
è
ø
for any x Î Be . Since h ( y ) = 0 , we obtain from (4.6) that
h(x) = h(x) - h (y)
£ G x - y £ Ge .
Therefore, estimates (4.3) and (4.4) imply that
T(x) - y £ e
for x Î Be ,
i.e. T maps the ball Be into itself. This mapping is contractive in Be . Indeed,
1- æ H (x , x ) + h(x ) - h(x ) ö ,
£ -----1
2
1
2 ø
2G è
T ( x1 ) - T ( x2 )
where
H ( x1 , x 2 ) º G ( x1 ) - G ( x2 ) - F ( x1 ) + F ( x2 ) =
1
=
ò [ D G ( x + x ( x - x ) ) - D F ( x + x ( x - x ) ) ] dx ( x - x ) .
1
1
2
1
1
2
1
2
0
It follows from (4.5) that
--- G x1 - x2 .
H ( x1 , x2 ) £ 1
2
This equation and inequality (4.6) imply the estimate
--- x1 - x2 .
T ( x1 ) - T ( x 2 ) £ 3
4
Therefore, the mapping T has a unique fixed point in the ball Be = { x :
x - y £ e } . The lemma is proved.
Let the hypotheses of Theorems 4.1 and 4.2 hold. We assume that h < 1 in (4.1).
Then for any element g Î Wh ( f ) the following estimates hold:
g (x) £ M ,
D g (x) £ M ,
x ÎO ,
(4.7)
where M > 0 is a constant. In particular, these estimates are valid for the mapping f .
Lemma 4.2.
Let { yn : n Î Z } be a d -pseudotrajectory of the mapping f lying in L .
Then for any k ³ 1 the sequence { zn º yn k : n Î Z } is a d × Mk - 1 -pseudotrajectory of the mapping f k . Here Mk has the form
Mk = 1 + M + ¼ + M k ,
and M is a constant from (4.7).
k ³ 1,
M0 = 1 ,
(4.8)
H y p e r b o l i c i t y o f I n v a r i a n tA nS oe st o
s vf’osr LDe im
f fm
e rae on nt i aeb-lter aMj ea cptpoi rnigess
Proof.
Let us use induction to prove that
yn k + i - f i ( yn k ) £ d × Mi - 1 ,
1 £ i £ k.
(4.9)
Since { yn } is a d -pseudotrajectory, then it is evident that for i = 1 inequality
(4.9) is valid. Assume that equation (4.9) is valid for some i ³ 1 and prove estimate (4.9) for i + 1 :
ynk + i + 1 - f i + 1 ( ynk )
£
ynk + i + 1 - f ( yn k + i ) + f ( ynk + i ) - f ( f i ( ynk) ) .
With the help of (4.7) we obtain that
ynk + i + 1 - f i + 1 ( ynk )
£ d + M yn k + i - f i ( yn k)
£ d + M × d Mi - 1 = d × Mi .
Thus, Lemma 4.2 is proved.
Lemma 4.3.
Let { yn } be a d -pseudoorbit of the mapping f lying in L . Let { xn } be
a trajectory of the mapping g Î Wh ( f ) such that
yn k - xn k £ e ,
n ÎZ,
(4.10)
for some k ³ 1 . If
max ( e , d + h ) Mk £ D ,
(4.11)
yn - xn £ max ( e , d + h ) × Mk ,
(4.12)
then
where M k has the form (4.8).
Proof.
We first note that
yn k + 1 - xn k + 1 = yn k + 1 - g ( xn k )
£
£
yn k + 1 - f ( yn k ) + f ( yn k ) - g ( yn k ) + g ( yn k ) - g ( xn k ) .
Therefore, it is evident that
yn k + 1 - xn k + 1
£ d + h + M yn k - xn k
£ max ( e , d + h ) ( 1 + M ) . (4.13)
Here we use the estimate
1
g (y) - g (x)
£
ò D g ( y + x ( x - y ) ) dx ×
x -y
£ M x -y
0
which follows from (4.7) and holds when the segment connecting the points x
and y lies in O . Condition (4.11) guarantees the fulfillment of this property
at each stage of reasoning. If we repeat the arguments from the proof of (4.13),
then it is easy to complete the proof of (4.12) using induction as in Lemma 4.2.
Lemma 4.3 is proved.
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Lemma 4.4.
Let y Î L and x - y £ e . Assume that
max ( e , h ) ( 1 + ¼ + M k - 1 ) < D .
(4.14)
Then the estimates
6
f j(y) - f j(x) £ M j x - y ,
f j(y) - g j(x)
D f j(x) £ M j ,
(4.15)
£ max ( e , h ) ( 1 + ¼ + M j ) ,
(4.16)
£ h ( 1 + ¼ + M j -1 )
(4.17)
f j(x) - g j(x)
are valid for j = 1 , 2 , ¼ , k and for every mapping g Î Wh ( f ) .
Proof.
As above, let us use induction. If j = 1 , it is evident that equations (4.15)–
(4.17) hold. The transition from j to j + 1 in (4.15) is evident. Let us consider
estimate (4.16):
f j + 1 ( y ) - g j + 1 ( x ) = æ f ( f j ( y ) ) - f ( g j ( x ) )ö + f ( g j ( x ) ) - g ( g j ( x ) ) .
è
ø
(4.18)
Condition (4.14) and the induction assumption give us that g j ( x ) lies in the ball
with the centre at the point f j ( y ) Î L lying in O . Therefore, it follows from
(4.18) that
f j + 1 ( y ) - g j + 1( x )
£ M f j ( y ) - g j ( x ) + h £ max ( e , h ) ( 1 + ¼ + M j +1 ) .
The transition from j to j + 1 in (4.17) can be made in a similar way. Lemma 4.4
is proved.
Lemma 4.5.
There exists D ¢ £ D such that the equations
ì
ü
sup í f k ( x ) - g k( x ) : x Î O ¢ý £ rk ( h )
î
þ
(4.19)
ì
ü
sup í ( D f k ) ( x ) - ( D g k ) ( x ) : x Î O ¢ý £ rk ( h )
î
þ
(4.20)
and
are valid in the D ¢ -vicinity O ¢ of the set L for any function
g Î Wh ( f ) . Here rk ( h ) ® 0 as h ® 0 .
The proof follows from the definition of the class of functions Wh ( f ) and estimates
(4.7) and (4.17).
H y p e r b o l i c i t y o f I n v a r i a n tA nS oe st o
s vf’osr LDe im
f fm
e rae on nt i aeb-lter aMj ea cptpoi rnigess
Let us also introduce the values
ì
w k ( š ) = sup í D f k ( y ) - D f k ( x ) : y Î L ,
î
ü
x - y £ šý
þ
(4.21)
ü
x - y £ šý .
þ
(4.22)
and
ì
w ( š ) = sup í P ( y ) - P ( x ) ,
î
x, y Î L ,
The requirement of the uniform continuity of the derivative D f ( x ) (see the hypotheses of Theorem 4.1) and the projectors P ( x ) (see the hyperbolicity definition)
enables us to state that
wk ( š ) ® 0 ,
w ( š ) ® 0 as š ® 0 .
(4.23)
Let { yn } be a d -pseudotrajectory of the mapping f lying in L . Then due to
Lemma 4.2 the sequence { yn = yn k : n Î Z } is a d Mk - 1-pseudotrajectory of the
mapping f k . Let us consider the mappings F ( x ) and G ( x ) in the space
¥
lX º l ¥( Z , X ) (for the definition see Section 2) given by the equalities
[ F ( x ) ] n = yn + xn - f k ( yn - 1 + xn - 1 ) ,
(4.24)
[ G ( x ) ] n = yn + xn - g k ( yn - 1 + xn - 1) ,
(4.25)
¥
where x = { xn : n Î Z } is an element from lX . Thus, the construction of e -trajectories of the mapping f k and g k corresponding to the sequence { yn } is reduced to
solving of the equations
F ( x ) = 0 and G ( x ) = 0
in the ball { x : x ¥ £ e } . Let us show that for k large enough Lemma 4.1 can be
lX
applied to the mappings F and G . Let us start with the mapping F .
Lemma 4.6.
¥
The function F is a C 1 -smooth mapping in lX with the properties
(4.26)
F ( 0 ) £ dMk - 1 ,
D F ( x ) - D F ( 0 ) £ wk ( e ) ,
x
¥
lX
£ e.
(4.27)
Proof.
Estimate (4.26) follows from the fact that { yn } is a d Mk - 1 -pseudotrajectory. Then it is evident that
[ D F ( x ) h ] n = hn - D f k ( yn - 1 + xn - 1 ) hn - 1 ,
(4.28)
¥
where x = { xn : n Î Z } and h = { hn : n Î Z } lie in lX
. Therefore, simple calculations and equation (4.21) give us (4.27).
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In order to deduce relations (4.2) and (4.3) from inequalities (4.26) and (4.27) for
y = 0 , we use Theorem 2.2. Consider the operator L = D F ( 0 ) . It is clear that
6
[ L h ] n = hn - D f k ( yn - 1 ) hn - 1 ,
h = { hn } .
Let us show that equations (2.9)–(2.11) are valid for An = D f k ( yn - 1 ) and Qn =
= P ( yn - 1 ) and then estimate the corresponding constants. Property (2.9) follows
from the hyperbolicity definition. Equations (3.3) and (4.15) imply that
An Qn £ K q k
and
An £ M k .
Further, the permutability property (3.2) gives us that
Qn + 1 An ( 1 - Qn ) = [ Qn + 1 An - An Qn ] ( 1 - Qn ) =
= Qn + 1 - P ( f k ( yn - 1 ) ) An ( 1 - Qn ) .
Hence (see (4.22)),
Qn + 1 An ( 1 - Qn )
£ w ( d Mk - 1 ) × An × 1 - Qn
£ w ( d Mk - 1 ) M k × K .
Similarly, we find that
( 1 - Qn + 1 ) An Qn
£ w ( d Mk -1 ) M k × K .
The operator
Jn = Qn + 1 P ( f k ( yn - 1 ) ) - ( 1 - Qn + 1 ) ( 1 - P ( f k ( yn - 1 ) ) )
is invertible if (see Lemma 3.1)
Qn + 1 - P ( f k ( yn - 1 ) )
1- .
£ w ( d Mk - 1 ) < -----2K
Moreover,
Jn-1
£ ( 1 - 2 K w ( d Mk - 1 ) ) -1 < 2 ,
provided 2 K w ( d Mk - 1 ) < 1 ¤ 2 . Due to the hyperbolicity of the set L , the operator
A n is an invertible mapping from ( 1 - Qn ) X into ( 1 - P ( f k ( yn - 1 ) ) ) X . Therefore,
since
( 1 - Qn + 1 ) An ( 1 - Qn ) = Jn An ( 1 - Qn ) ,
the operator ( 1 - Q n + 1 ) An ( 1 - Qn ) is invertible as a mapping from ( 1 - Qn ) X into
( 1 - Qn + 1 ) X . Moreover, by virtue of (3.4) we have that
[ ( 1 - Qn + 1 ) An ] -1(1 - Qn + 1) = [A n ( 1 - Qn ) ] -1 × Jn-1(1 - Qn + 1)
£ 2K2qk .
Thus, under the conditions
4 K w ( d Mk - 1 ) < 1 ,
8 KM k w ( d Mk - 1 ) £ 1 ,
16 K 3 q k £ 1 ,
(4.29)
Theorem 2.2 implies that the operator L = D F ( 0 ) is invertible and L-1 £ 2 K + 1 .
Let us fix some e > 0 . If
H y p e r b o l i c i t y o f I n v a r i a n tA nS oe st o
s vf’osr LDe im
f fm
e rae on nt i aeb-lter aMj ea cptpoi rnigess
--- ( 4 K + 2 ) -1 × e ,
d Mk - 1 £ 1
w k ( e ) £ ( 4 K + 2 ) -1 ,
(4.30)
2
then by Lemma 4.6 relations (4.2) and (4.3) hold with y = 0 , e = e , and q = 1 ¤ 2 . If
--- min ( e , 1 ) ( 4 K + 2 ) -1 ,
rk ( h ) £ 1
2
(4.31)
then equations (4.4) and (4.5) also hold with y = 0 , e = e , and q = 1 ¤ 2 . Hence,
under conditions (4.29)–(4.31) there exists a unique solution to equation G ( x ) = 0
possessing the property x l ¥ £ e . This means that for any d -pseudoorbit { yn :
n Î Z } (lying in L ) of the mapping f there exists a unique trajectory { zn : n Î Z }
of the mapping g Î Wh ( f ) such that
zn k - yn k £ e ,
n ÎZ,
provided conditions (4.29)–(4.31) hold. Therefore, under the additional condition
( e + d + h ) Mk £ D
and due to Lemma 4.3 we get
yn - zn £ ( e + d + h ) Mk ,
n ÎZ.
These properties are sufficient for the completion of the proof of Theorem 4.2.
Let us fix k such that 16 K 3 q k £ 1 . We choose e 0 £ D ¢ £ D ( D ¢ is defined
in Lemma 4.5) such that
w k ( e ) £ ( 4 K + 2 ) -1
for all
e0
-.
e £ ---------2 Mk
Let us fix an arbitrary e Î ( 0 , e 0 ] and take e = e [ 2 Mk ] -1 . Now we choose
d = d ( e ) and h = h ( e ) such that the following conditions hold:
4 K w ( d Mk - 1 ) < 1 ,
--- e ( 2 K + 1 ) -1 ,
d Mk - 1 £ 1
4
8 K M k w ( d Mk - 1 ) £ 1 ,
--- min ( e , 1 ) ( 2 K + 1 ) -1 ,
rk ( h ) £ 1
4
2 ( d + h ) Mk £ e .
It is clear that under such a choice of d and h any d -pseudoorbit (from L) of the
mapping f has a unique e -trajectory of the mapping g . Thus, Theorem 4.2 is proved.
E x e r c i s e 4.1 Let the hypotheses of Theorem 4.1 hold. Show that there
exist D > 0 and d > 0 such that for any two trajectories { xn : n ÎZ }
and { yn : n Î Z } of a dynamical system ( X , f ) the conditions
dist ( xn , L ) < D ,
dist ( yn , L ) < D ,
sup xn - yn £ d
n
imply that xn º yn , n Î Z . In other words, any two trajectories of
the system ( X , f ) that are close to a hyperbolic invariant set cannot
remain arbitrarily close to each other all the time.
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Homoclinic Chaos in Infinite-Dimensional Systems
E x e r c i s e 4.2 Show that Theorem 4.1 admits the following strengthening:
if the hypotheses of Theorem 4.1 hold, then there exists e 0 > 0 such
that for every e Î ( 0 , e0 ) there exists d = d ( e ) with the property
that for any d -pseudoorbit { yn } such that dist ( yn , L ) < d there
exists a unique e -trajectory.
E x e r c i s e 4.3 Prove the analogue of the assertion of Exercise 4.2 for Theorem 4.2.
E x e r c i s e 4.4 Let L = { xn : n Î Z } be a periodic orbit of the mapping f ,
i.e. f k ( xn ) º xn + k = xn for all n Î Z and for some k ³ 1 . Assume
that the hypotheses of Theorem 4.2 hold. Then for h > 0 small
enough every mapping g Î Wh ( f ) possesses a periodic trajectory
of the period k .
§ 5
Birkhoff-Smale Theorem
One of the most interesting corollaries of Anosov’s lemma is the Birkhoff-Smale theorem that provides conditions under which the chaotic dynamics is observed in
a discrete dynamical system ( X , f ) . We remind (see Section 1) that by definition
the possibility of chaotic dynamics means that there exists an invariant set Y in the
space X such that the restriction of some degree f k of the mapping f on Y is topologically equivalent to the Bernoulli shift S in the space S m of two-sided infinite
sequences of m symbols.
Theorem 5.1.
Let f be a continuously differentiable mapping of a Banach space X
into itself. Let x0 Î X be a hyperbolic fixed point of f and let { yn : n Î Z }
be a homoclinic trajectory of the mapping f that does not coincide with
x0 , i.e.
f ( x0 ) = x0 ;
f ( yn ) = yn + 1 ,
yn ¹ x0 ,
n ÎZ;
yn ® x0 , n ® ± ¥ .
Assume that the trajectory { yn : n Î Z } is transversal, i.e. the set
L = { x0 } È { yn : n Î Z }
is hyperbolic with respect to f and there exists a vicinity O of the set L
such that f ( x ) and D f ( x ) are bounded and uniformly continuous on the
closure O . By Wh ( f ) we denote a set of continuously differentiable mappings g of the space X into itself such that
f (x) - g (x) £ h ,
D f (x) - D g (x) £ h ,
x ÎO .
Birkhoff-Smale Theorem
Then there exists h > 0 such that for any mapping g Î Wh ( f ) and for any
m ³ 2 there exist a natural number l and a continuous mapping j of the
space Sm into a compact subset Y º j ( S m ) in X such that
a) Y = j ( S m ) is strictly invariant with respect to g l , i.e. g l ( Y ) = Y ;
b) if a = ( ¼ a-1 , a 0 , a1 , ¼ ) and a ¢ = ( ¼ a-¢ 1 , a 0¢ , a1¢ , ¼ ) are elements
of S m such that ai ¹ ai¢ for some i ³ 0 , then j ( a ) ¹ j ( a ¢) ;
c) the restriction of g l on Y is topologically conjugate to the Bernoulli
shift S in S m , i.e.
g l(j (a) ) = j (S a) ,
a Î Sm .
Moreover, if in addition we assume that for the mapping g there exists
e0 > 0 such that for any two trajectories { xn : n Î Z } and { xn : n Î Z }
(of the mapping g ) lying in the e 0 -vicinity of the set L the condition
xn = xn for some n 0 Î Z implies that xn = xn for all n Î Z , then the map0
0
ping j is a homeomorphism.
The proof of this theorem is based on Anosov’s lemma and mostly follows the standard scheme (see, e.g., [4]) used in the finite-dimensional case. The only difficulty
arising in the infinite-dimensional case is the proof of the continuity of the mapping
j . It can be overcome with the help of the lemma presented below which is borrowed from the thesis by Jürgen Kalkbrenner (Augsburg, 1994) in fact.
It should also be noted that the condition under which j is a homeomorphism
holds if the mapping g does not “glue” the points in some vicinity of the set L , i.e.
the equality g ( x ) = g ( x ) implies x = x .
Lemma 5.1.
Let the hypotheses of Theorem 5.1 hold. Let us introduce the notation
Jn º Jn ( k0 , m ) = { k Î Z : k - k0 £ m n } ,
where k0 Î Z and m , n Î N . Let z = { zn : n Î Jn } be a segment (lying
in L ) of a d - pseudoorbit of the mapping f :
zn Î L ,
zn + 1 - f ( zn ) £ d ,
n , n + 1 Î Jn .
(5.1)
Assume that x = { xn : n Î Jn } and x = { xn : n Î Jn } are segments of orbits of the mapping g Î Wh ( f ) :
g ( xn ) = xn + 1 ,
g ( xn ) = xn + 1 ,
n , n + 1 Î Jn ,
(5.2)
such that
zn - xn £ e ,
zn - xn £ e .
(5.3)
Then there exist d , h , e > 0 , and m Î N such that conditions (5.1)–
(5.3) imply the inequality
xk - xk £ 2 1 - n e .
0
0
(5.4)
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Proof.
It follows from (5.2) that
xk + m - xk + m = ( D f m ) ( zk ) ( xk - xk ) + R k ,
0
0
0
0
0
0
(5.5)
where
Rk = g m ( xk ) - g m ( xk ) - ( D f m ) ( zk ) ( xk - xk ) .
0
0
0
0
0
0
Since the set L is hyperbolic with respect to f , there exists a family of projectors { P ( x ) : x Î L } for which equations (3.2)–(3.4) are valid. Therefore,
( 1 - P ( f m ( zk ) ) ) ( xk
0
0 +m
- xk
0 +m
)=
= ( D f m )(zk ) ( 1 - P ( zk ) ) ( xk - xk ) + ( 1 - P ( f m ( zk ) ) ) Rk .
0
0
0
0
0
0
It means that
( 1 - P ( zk ) ) ( xk - xk ) =
0
0
= [ ( D f m )(zk ) ]
0
-1
0
[ 1 - P ( f m ( zk ) ) ] [ xk
0
0 +m
- xk
0 +m
- Rk ] .
0
Consequently, equation (3.4) implies that
£ K q m ( xk
( 1 - P ( zk ) ) ( xk - xk )
0
0
0
0 +m
- xk
+ Rk ) .
0 +m
0
Let us estimate the value Rk . It can be rewritten in the form
0
1
Rk =
0
ò [ (D g ) (x x
m
- D f m ( zk ) ] dx × ( xk - xk ) .
k0 + ( 1 - x ) xk0 )
0
0
0
0
It follows from (5.3) that x xk + ( 1 - x ) xk - z k
0
0
0
and (4.21), for e > 0 small enough we obtain that
£ e . Hence, using (4.20)
£ ( r m ( h ) + wm ( e ) ) xk - xk ,
Rk
0
0
(5.6)
0
where rm ( h ) ® 0 as h ® 0 and wm( e ) ® 0 as e ® 0 . Therefore,
( 1 - P ( zk ) ) ( xk - xk )
0
0
0
£ K q m ( xk
0 +m
- xk
0 +m
+ xk - xk ) ,
0
(5.7)
0
provided rm ( h ) + wm ( e ) £ 1 . Further, we substitute the value k0 - m for k0
in (5.5) to obtain that
xk - xk = D f m ( zk - m ) ( xk - m - xk - m ) + R k - m .
0
0
0
0
0
0
Therefore, using (3.2) we find that
P ( f m ( z k - m ) ) ( xk - xk ) =
0
0
0
= D f m ( zk
0 -m
) P ( zk
0 -m
) ( xk
0 -m
- xk
0 -m
) + P ( f m ( zk
0 -m
) ) Rk
0 -m
.
Birkhoff-Smale Theorem
Hence, equations (3.3) and (5.6) with k0 - m instead of k0 give us that
P ( f m ( z k - m ) ) ( xk - xk )
0
0
0
£
£ K { q m + ( rm ( h ) + wm ( e ) ) } xk
0 -m
- xk
0 -m
.
(5.8)
Since
zk - f m ( zk - m ) =
0
0
m -1
ì
å íî f (z
j=0
j
k0 - j )
ü
- f j ( f ( zk
)) ý
0 - j -1
þ
,
it follows from (4.15) and (5.1) that
zk - f m ( zk - m )
0
0
m -1
£ d
åM
j
£ d × m (1 + M m)
j=0
for d small enough. Therefore,
P ( zk ) - P ( f m ( z k - m ) )
0
0
£ w (d × m (1 + M m) ) º w (d , m) ,
where w ( x ) ® 0 as x ® 0 (cf. (4.22)). Consequently, estimate (5.8) implies that
P ( zk ) ( xk - x k )
0
0
0
£
£ K { q m + rm ( h ) + wm ( e ) + K -1 × w ( d , m ) } xk
0 -m
- xk
0 -m
.
(5.9)
It is evident that estimates (5.7) and (5.9) enable us to choose the parameters
m , h , e , and d such that
xk - xk
0
0
--- max { xk - xk : k Î J1 ( k0 , m ) } .
£ 1
2
Using this inequality with k instead of k0 we obtain that
--- max { xn - xn : n Î J2 ( k0 , m ) }
xk - xk £ 1
2
for all k Î J1 ( k0 , m ) . Therefore,
--- max { xn - xn : n Î J2 ( k0 , m ) } .
xk - xk £ 1
4
0
0
If we continue to argue like that, then we find that
xk - xk
0
0
£ 2 -n max { xn - xn : n Î Jn ( k0 , m ) } .
Since xn - xn £ xn - zn + xn - zn , this and estimate (5.3) imply (5.4).
Lemma 5.1 is proved.
Proof of Theorem 5.1.
Let p1 , p2 , ¼ , pm - 1 be distinct integers. Let us choose and fix the parameters e , h , d > 0 and the integer m > 0 such that (i) Theorem 4.2 and Lemma 5.1
can be applied to the hyperbolic set L = { x0 } È { yn : n Î Z } and (ii)
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ì
--- min í yp - yp , yp - x0 :
e < 1
2
j
i
î i
ü
i, j = 1, ¼, m -1, i ¹ j ý .
þ
(5.10)
Assume that N is such that
d- for n = N + p + 1 and n = p - N
yn - x0 < -(5.11)
i
i
2
for all i = 1 , 2 , ¼ , m - 1 . Let us consider the segments Ci of the orbits of the mapping f of the form
Ci = ( yp - N , ¼ , yp , ¼ , yp + N ) ,
i
i
i
i = 1, 2, ¼, m -1 ,
Cm = ( x0 , x0 , ¼ , x0 ) .
The length of every such segment is 2 N + 1 . Let a = ( ¼ a-1 a0 a1 ¼ ) Î Sm . Let us
consider a sequence of elements ga made up of the segments Ci by the formula
ga = ( ¼ Ca Ca Ca ¼ ) .
-1
0
(5.12)
1
It is clear that g a Î L and by virtue of (5.11) g a is a d -pseudoorbit of the mapping
f . Therefore, due to Theorem 4.2 there exists a unique trajectory { wn º wn ( a ) :
n Î Z } of the mapping g such that
wn ( N , i , j ) - z i j ( a )
£ e,
(5.13)
where n ( N , i , j ) = N + ( i - 1 )(2 N + 1) + j and z i j ( a ) is the j -th element of the
segment Cai , i Î Z , j = 1 , 2 , ¼ , 2 N + 1 . Let us define the mapping j from S m
into X by the formula
j ( a ) = w0 ,
(5.14)
where w0 is the zeroth element of the trajectory { wn } . Since the trajectory { wn }
possessing the property (5.13) is uniquely defined, equation (5.14) defines a mapping from S m into X .
If we substitute i + 1 for i in (5.13) and use the equations
wn ( N , i + 1 , j ) = wn ( N , i , j ) + 2 N + 1 = g 2 N + 1 æ wn ( N , i , j )ö ,
è
ø
we obtain that
g 2 N + 1 ( wn ( N , i , j ) ) - zi + 1 , j ( a )
£ e
for all i Î Z and j = 1 , 2 , ¼ , 2 N + 1 . Therefore, the equality zi + 1 , j ( a ) = zi j ( S a )
with S being the Bernoulli shift in S m leads us to the equation
g 2 N +1 ( wn ( N , i , j ) ) - z i + 1 , j ( S a )
£ e.
Consequently, the uniqueness property of the e -trajectory in Theorem 4.2 gives us
the equation
wn ( S a ) = g 2 N + 1 ( wn ( a ) ) ,
n ÎZ.
Birkhoff-Smale Theorem
This implies that
j ( S a ) = g 2 N + 1(j ( a )) ,
a Î Sm ,
(5.15)
i.e. property (c) is valid for l = 2 N + 1 . It follows from (5.15) that
g 2 N + 1 ( j ( S -1 a ) ) = j ( a ) .
Therefore, the set Y = j ( Sm ) is strictly invariant with respect to g 2 N + 1 . Thus, assertion (a) is proved.
Let us prove the continuity of the mapping j . Assume that the sequence of elements a ( s ) = ( ¼ , a-( s1) , a0( s ) , a1( s ) , ¼ ) of S m tends to a = ( ¼ , a-1 , a0 , a1 , ¼ ) Î
Î S m as s ® + ¥ . This means (see Exercise 1.4) that for any M Î N there exists
s0 = s 0 ( M ) such that
ai( s ) = ai
for
i £ M , s ³ s0 .
(5.16)
(s)
{ zk }
Assume that g ( s ) =
and g = { z k } are d -pseudoorbits in L constructed according to (5.12) for the symbols a ( s ) and a , respectively. Equation (5.16) implies
that z k( s ) = z k for k £ M ( 2 N + 1 ) . Let { wn } and { wn( s ) } be e -trajectories corresponding to g and g ( s ) , respectively. Lemma 5.1 gives us that
(s)
w0 - w0 £ 2 1 - n e ,
(5.17)
provided M ( 2 N +1 ) ³ nm , i.e. for any n ÎN equations (5.17) is valid for s ³ s0(n , M) .
This means that
j ( a ( s ) ) - j ( a ) º w0( s ) - w0 ® 0
as s ® + ¥ . Thus, the mapping j is continuous and Y = j ( Sm ) is a compact strictly invariant set with respect to g 2 N + 1 .
Let us now prove nontriviality property (b) of the mapping j . Let a , a ¢ Î S m
be such that ai ¹ ai¢ for some i ³ 0 . Let { wn } and { w n¢ } be e -trajectories corresponding to the symbols a and a ¢ , respectively. Then
wn ( N , i , N + 1 ) - wn¢ ( N , i , N + 1 ) ³
z i , N + 1 ( a ) - z i , N + 1 ( a ¢) -
- wn ( N , i , N + 1 ) - zi , N + 1 ( a ) - wn¢ ( N , i , N + 1 ) - zi , N + 1 ( a ¢) .
Therefore, it follows both from (5.13) and the definition of the elements z i j ( a ) that
w( 2 N + 1 ) i - w (¢ 2 N + 1 ) i ³ yq - yq ¢ - 2 e ,
i
i
where qi = pa and q i¢ = pa ¢ . We apply (5.10) to obtain that
i
i
w( 2 N + 1 ) i - w (¢ 2N + 1 ) i > 0 .
Therefore, if i ³ 0 , then
g ( 2 N + 1 ) i ( w0 ) ¹ g ( 2 N + 1 ) i ( w 0¢ ) .
Hence, j ( a ) º w 0 ¹ w 0¢ º j ( a ¢) . This completes the proof of assertion (b).
(5.18)
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Homoclinic Chaos in Infinite-Dimensional Systems
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If the trajectories of the mapping g cannot be “glued” (see the hypotheses of
Theorem 5.1), then for some i Î Z equation (5.18) gives us that w0 ¹ w 0¢ , i.e.
j ( a ) ¹ j ( a ¢) if a ¹ a ¢ . Thus, the mapping j is injective in this case. Since S m
is a compact metric space, then the injectivity and continuity of j imply that j is
a homeomorphism from S m onto j ( S m ) . Theorem 5.1 is proved.
6
It should be noted that equations (5.13) and (5.14) imply that the set Y = j ( S m )
lies in the e -vicinity of the hyperbolic set L . Therewith, the values h and l involved in the statement of the theorem depend on e and one can state that for any
vicinity U of the set L there exist h and l such that the conclusions of Theorem 5.1 are valid and j ( Sm ) Ì U . It is also clear that the set Y = j ( S m ) is not
uniquely determined.
E x e r c i s e 5.1 Assume that g = f in Theorem 5.1. Prove that the mapping j
can be constructed such that j ( Sm ) É { x0 } È { yn l : n ³ 0 } , where
{ yn : n Î Z } is a homoclinic orbit of the mapping f .
E x e r c i s e 5.2 Prove the Birkhoff theorem: if the hypotheses of Theorem 5.1
hold, then for any e > 0 small enough there exist h > 0 and l Î N
such that for every mapping g Î Wh ( f ) there exist periodic trajectories of the mapping g l of any minimal period in the e -vicinity
of the set L .
E x e r c i s e 5.3 Use Theorem 1.1 to describe all the possible types of behaviour of the trajectories of the mapping g on a set
l
W=
Èg
k=1
k (j (S
m) ) .
In conclusion, it should be noted that different infinite-dimensional versions of Anosov’s lemma and the Birkhoff-Smale theorem have been considered by many authors
(see, e.g., [6], [10], [11], [12], and the references therein).
§ 6
Possibility of Chaos in the Problem
of Nonlinear Oscillations of a Plate
In this section the Birkhoff-Smale theorem is applied to prove the existence of chaotic
regimes in the problem of nonlinear plate oscillations subjected to a periodic load.
The results presented here are close to the assertions proved in [13]. However, the
methods used differ from those in [13].
Possibility of Chaos in the Problem of Nonlinear Oscillations of a Plate
Let us remind the statement of the problem. We consider its abstract version
as in Chapter 4. Let H be a separable Hilbert space and let A be a positive operator
with discrete spectrum in H , i.e. there exists an orthonormalized basis { e k } in H
such that
Aek = l k e k ,
0 < l1 £ l 2 £ ¼ ,
lim l n = ¥ .
n®¥
The following problem is considered:
ì u·· + g u· + A2 u + ( š A1/ 2 u 2 - G ) A u + L u = h cos w t ,
(6.1)
ï
í
ïu
(6.2)
= u 0 Î F1 = D ( A ) , u· t = 0 = u1 Î H .
î t=0
Here g , š , G , and w are positive parameters, h is an element of the space H , L
is a linear operator in H subordinate to A , i.e.
L u £ K Au ,
(6.3)
where K is a constant. The problem of the form (6.1) and (6.2) was studied in Chapter 4 in details (nonlinearity of a more general type was considered there). The results of Section 4.3 imply that problem (6.1) and (6.2) is uniquely solvable in the
class of functions
W+ = C ( R+ , D ( A ) ) Ç C 1 ( R+ , H ) .
(6.4)
Moreover, one can prove (cf. Exercise 4.3.9) that Cauchy problem (6.1) and (6.2)
is uniquely solvable on the whole time axis, i.e. in the class
W = C ( R , D ( A) ) Ç C 1 ( R , H ) .
This fact as well as the continuous dependence of solutions on the initial conditions
(see (4.3.20)) enables us to state that the monodromy operator G acting in H =
= D ( A) ´ H according to the formula
p-ö , u· æ 2
p-ö ö
----------G ( u0 ; u1 ) = æ u æ 2
è è wø
è w øø
(6.5)
is a homeomorphism of the space H (see Exercise 4.3.11). Here u ( t ) is a solution
to problem (6.1) and (6.2)
The aim of this section is to prove the fact that under some conditions on L and
h chaotic dynamics is observed in the discrete dynamical system ( H , G ) for some
set of parameters g , š , G , and w .
Lemma 6.1.
The mapping G defined by equality (6.5) is a diffeomorphism of the
space H = D ( A) ´ H .
Proof.
We use the method applied to prove Lemma 4.7.3. Let u1 ( t ) be a solution
to problem (6.1) and (6.2) with the initial conditions y = ( u0 , u1 ) Î H and let
397
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Homoclinic Chaos in Infinite-Dimensional Systems
u2 ( t ) be a solution to it with the initial condions y + z º ( u0 + z 0 , u1 + z1 ) ÎH .
Let us consider a linearization of problem (6.1) and (6.2) along the solution u1( t ) :
·· ( t ) + g w· ( t ) + A2 w + ( š A1/ 2 u ( t ) 2 - G ) Aw +
ìw
1
ï
ï
1
2
1
2
/
/
+ 2 š ( A u1 ( t ) , A w ( t ) ) Au1 ( t ) + L w = 0 ,
í
ï
ïw
= z0 , w· t = 0 = z1 .
î t=0
6
(6.6)
(6.7)
As in the proof of Theorem 4.2.1, it is easy to find that problem (6.6) and (6.7) is
uniquely solvable in the class of functions (6.4). Let v ( t) = u2 ( t) - u1 ( t) - w ( t) .
It is evident that v ( t ) is a weak solution to problem
ì v·· + g v· ( t ) + A2 v ( t ) = F ( t ) º F ( u ( t ) , u ( t ) , w ( t ) ) ,
1
2
ï
í
ïv
= 0 , v· t = 0 = 0 ,
î t=0
(6.8)
(6.9)
where
F ( u1 , u2 , w ) = -æ š A1/ 2 u2 2 - Gö Au 2 + æ š A1/ 2 u1 2 - Gö Au1 +
è
ø
è
ø
+ æ š A1/ 2 u1
è
2
- Gö Aw + 2 š ( Au1 , w ) Au1 - L ( u 2 - u1 - w ) .
ø
A simple calculation shows that
F ( u1 , u2 , w ) = F1 ( u1 , u2 , w ) + F2 ( u1 , u 2 , w ) º F1 + F2 ,
where
F1 = -æ š A1/ 2 u2 2 - Gö Av - L v - 2 š ( Au1 , v ) A ( u1 + w ) ,
è
ø
F2 = -š A1/ 2 ( u1 - u 2 ) 2 A ( u1 + w ) - 2 š ( Au1 , w ) Aw .
We assume that y
mates
H
£ R and z
H
£ 1 . In this case (see Section 4.3) the esti-
Au j ( t ) £ CR , T ,
A ( u1 ( t ) - u2 ( t ) ) £ CR , T z
Aw ( t ) £ C R , T z
H,
H,
(6.10)
are valid on any segment [ 0 , T ] . Here CR , T is a constant. Therefore,
F1 £ C1 Av ,
F2
£ C2 z
2
H
,
t Î [ 0, T] ,
where C1 and C2 are constants depending on R and T. Hence,
4
F ( t ) 2 £ C1 Av 2 + C2 z H
,
t Î [ 0, T ] .
Possibility of Chaos in the Problem of Nonlinear Oscillations of a Plate
Therefore, the energy equation
t
1
--- æ v· ( t ) 2 + Av ( t ) 2ö + g
ø
2è
ò
t
v· ( t ) 2 dt =
0
ò(F(t) , v· (t) ) dt
(6.11)
0
for problem (6.8) and (6.9) leads us to the estimate
t
v· ( t ) 2 + Av ( t ) 2 £
òæèC
1
Av ( t ) 2 + C2 z 4ö dt ,
ø
t Î [0, T ] .
0
Using Gronwall’s lemma we find that
v· ( t ) 2 + Av ( t ) 2 £ C z 4 ,
t Î [0, T ] ,
where the constant C depends on T and R . This estimate implies that the mapping
p-ö ; w· æ -----2 p-ö ö
-----G ¢ : z º ( z0 ; z1 ) ® æ w æ 2
è è wø
è w øø
(6.12)
is a Frechét derivative of the mapping G defined by equality (6.5). Here w ( t )
is a solution to problem (6.6) and (6.7). It follows from (6.10) and (6.6) that G ¢
is a continuous linear mapping of H into itself. Using (6.10) it is also easy to see
that G ¢ = G ¢ [ u 0 , u1 ] continuously depends on y = ( u0 ; u1 ) with respect to
the operator norm. Lemma 6.1 is proved.
Further we will also need the following assertion.
Lemma 6.2.
Let Gj be the monodromy operator of problem (6.1) and (6.2) with
L = Lj and h = hj , j = 1 , 2 . Assume that for L = Lj equation (6.3) is valid and hj Î H , j = 1 , 2 . Moreover, assume that
Lj A-1 £ r ,
hj £ r ,
j = 1, 2 .
Then the estimates
sup G1 ( y ) - G2 ( y )
£ C æ ( L1 - L2 ) A-1 + h1 - h2 ö
è
ø
(6.13)
sup G1¢ ( y ) - G2¢ ( y )
£ C æ ( L1 - L2 ) A-1 + h1 - h 2 ö
è
ø
(6.14)
y Î BR
and
y Î BR
are valid. Here BR is a ball of the radius R in H = D ( A) ´ H , R > 0
is an arbitrary number while the constant C depends on R and r but
does not depend on the parameters w , g , š ³ 0 , and G provided they
vary in bounded sets.
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Proof.
Let uj ( t ) be a solution to problem (6.1) and (6.2) with L = Lj and h = hj ,
j = 1 , 2 . It is evident (see Section 4.3) that
6
Therefore, it is easy to find that the difference u ( t ) = u1 ( t ) - u2 ( t ) satisfies
the equation
Au j ( t ) £ C º C ( R , r , T ) ,
t Î [0, T ] .
(6.15)
ì u·· + g u + A2 u = F ( t , u , u ) ,
1
2
ï
í
ï u t = 0 = 0 , u· t = 0 = 0 ,
î
where the function F ( t , u1 , u2 ) can be estimated as follows:
F ( t , u1 , u2 ) £ C1 Au + C2 æ ( L1 - L2 ) A-1 + h1 - h2 ö .
è
ø
As in the proof of Lemma 6.1, we now use energy equality (6.11) and Gronwall’s
lemma to obtain the estimate
u· ( t ) 2 + Au ( t ) 2 £ C æ ( L1 - L2 ) A-1 2 + h1 - h 2 2ö .
è
ø
(6.16)
This implies inequality (6.13). Estimate (6.14) can be obtained in a similar way.
In its proof equations (6.12), (6.15), and (6.16) are used. We suggest the reader
to carry out the corresponding reasonings himself/herself. Lemma 6.2 is proved.
Let us now prove that there exist an operator L and a vector h such that the corresponding mapping G possesses a hyperbolic homoclinic trajectory. To do that, we use
the following well-known result (see, e.g., [1], [13], as well as Section 7) related to the
Duffing equation.
Theorem 6.1.
Let g : R 2 ® R 2 be a monodromy operator corresponding to the Duffing
equation
x·· - b x + a x 3 = e ( f × cos w t - d x· ) ,
(6.17)
i.e. the mapping of the plane R 2 into itself acting according to the formula
p-ö ; x· æ 2
p-ö ö ,
----------g ( x 0 , x1 ) = æ x æ 2
è è wø
è w øø
(6.18)
where x ( t ) is a solution to equation (6.17) such that x ( 0 ) = x0 and x· ( 0 ) =
= x1 . All the parameters contained in (6.17) are assumed to be positive.
Let us also assume that
2 b3 / 2
pw
f > fcr º d × --------------------- × cosh æ ---------- ö .
è
3w 2 a
2 bø
(6.19)
Possibility of Chaos in the Problem of Nonlinear Oscillations of a Plate
Then there exists e0 > 0 such that for every e Î ( 0 , e0 ) the mapping g possesses a fixed point z and a homoclinic trajectory { yn : n Î Z } to it, yn ¹ z ,
therewith the set { z } È { yn : n Î Z } is hyperbolic.
Let Pk be the orthoprojector onto the one-dimensional subspace generated by the
eigenvector e k in H . We consider problem (6.1) and (6.2) with Lk = L ( 1 - Pk ) instead of L and h = hk × e k , where hk is a positive number. Then it is evident that
every solution to problem (6.1) and (6.2) with the initial conditions u 0 = c0 e k and
u1 = c1 e k has the form
u ( t ) = x ( t ) ek ,
where x ( t ) is a solution to the Duffing equation
x·· + g x· - l k ( G - l k ) x + š l2k x 3 = h k cos w t
(6.20)
with the initial conditions x ( 0 ) = c 0 and x· ( 0 ) = c1 . In particular, this means that
the two-dimensional subspace Lk = Lin { ( e k ; 0 ) , ( 0 ; e k ) } of the space H is strictly invariant with respect to the corresponding monodromy operator Gk while the restriction of Gk to L k coincides with the monodromy operator corresponding to the
Duffing equation (6.20). Therefore, if hk is small enough and the conditions
0 < lk < G ,
hk
2 lk ( G - lk ) æ G
1/2
pw
------ > ------------------------------ ------ - 1ö cosh ----------------------------------è
ø
g
lk
3w 2š
2 lk ( G - lk )
hold, then the mapping Gk possesses a hyperbolic invariant set
Lk = { z ek } È { yn × ek : n Î Z }
consisting of the fixed point ( z 0 ek ; z1 ek ) and its homoclinic trajectory
{ yn ek : n Î Z } , where yn = ( yn0 ; yn1 ) Î R 2 .
Thus, if w > 0 and for some k the condition 0 < l k < G holds, then there exists an
open set P in the space of parameters { g , hk } such that for every ( g , hk) ÎP the monodromy operator Gk corresponding to problem (6.1) and (6.2) with Lk = L ( 1 - Pk )
instead of L and h = hk e k possesses a hyperbolic set consisting of a fixed point and
a homoclinic trajectory. This fact as well as Lemmata 6.1 and 6.2 enables us to apply
the Birkhoff-Smale theorem and prove the following assertion.
Theorem 6.2.
Let w > 0 and let the condition 0 < lk < G hold for some k . Then there
exist m > 0 and an open set P in the metric space R + ´ H such that if
Lek l-k1 < m ,
(g, h) Î P ,
then some degree G l of the monodromy operator G of problem (6.1) and
(6.2) possesses a compact strictly invariant set Y ( G l Y = Y ) in the space H
in which the mapping G l is topologically conjugate to the Bernoulli shift of
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Homoclinic Chaos in Infinite-Dimensional Systems
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sequences of m symbols, i.e. there exists a homeomorphism j : S m ® Y
such that
6
G l (j (a) ) = j (S a) ,
a Î Sm .
E x e r c i s e 6.1 Prove that if the hypotheses of Theorem 6.2 hold, then equation (6.1) possesses an infinite number of periodic solutions with periods multiple to w .
E x e r c i s e 6.2 Apply Theorem 6.2 to the Berger approximation of the problem of nonlinear plate oscillations:
ì
ï
ï
ï
ï
í
ï
ï
ï
ï
î
æ
¶2 u
¶u
--------+ g ------- + D2 u - ç š
2
¶
t
¶t
è
ò Ñu ( x , t )
W
¶u
+ r --------- = h ( x ) cos w t ,
¶x 1
u ¶W = D u ¶W = 0 ,
§7
2
ö
dx - G÷ Du +
ø
x = ( x1 , x2 ) Î W Ì R 2 ,
u t = 0 = u0 ( x ) ,
¶u
------¶t
t=0
t >0 ,
= u1 ( x ) .
On the Existence of Transversal
Homoclinic Trajectories
Undoubtedly, Theorem 6.1 on the existence of a transversal (hyperbolic) homoclinic
trajectory of the monodromy operator for the periodic perturbation of the Duffing
equation is the main fact which makes it possible to apply the Birkhoff-Smale
theorem and to prove the possibility of chaotic dynamics in the problem of plate
oscillations. In this connection, the question as to what kind of generic condition
guarantees the existence of a transversal homoclinic orbit of monodromy operators
generated by ordinary differential equations gains importance. Extensive literature
is devoted to this question (see, e.g., [1], [2] and the references therein). There are
several approaches to this problem. All of them enable us to construct systems with
transversal homoclinic trajectories as small perturbations of “simple” systems with
homoclinic (not transversal!) orbits. In some cases the corresponding conditions on
perturbations can be formulated in terms of the Melnikov function.
This section is devoted to the exposition and discussion of the results obtained
by K. Palmer [14]. These results help us to describe some classes of systems of ordinary differential equations which generate dynamical systems with transversal ho-
On the Existence of Transversal Homoclinic Trajectories
moclinic orbits. Such differential equations are obtained as periodic perturbations
of autonomous equations with homoclinic trajectories.
In the space R n let us consider a system of equations
x· ( t ) = g ( x ( t ) ) ,
x ( t ) Î Rn ,
(7.1)
where g : R n ® R n is a twice continuously differentiable mapping. Assume that the
Cauchy problem for equation (7.1) is uniquely solvable for any initial condition
x ( 0 ) = x0 . Let us also assume that there exist a fixed point z 0 ( g ( z0 ) =0 ) and
a trajectory z ( t ) ¹ z 0 homoclinic to z 0 , i.e. a solution to equation (7.1) such that
z ( t ) ® z 0 as t ® ± ¥ . Exercises 7.1 and 7.2 given below give us the examples of the
cases when these conditions hold. We remind that every second order equation x·· +
+ V ( x ) = 0 can be rewritten as a system of the form (7.1) if we take x1 = x and
x2 = x· .
E x e r c i s e 7.1
Consider the Duffing equation
x·· - b x + a x 3 = 0 ,
a, b > 0 .
·
Prove that the curve z ( t ) = (h ( t ) , h ( t ) ) Î R 2 is an orbit of the corresponding system (7.1) homoclinic to 0 . Here h ( t ) = 2 b ¤ a sech b t .
E x e r c i s e 7.2 Assume that for a function U ( x ) Î C 3 ( R ) there exist a number E and a pair of points a < b such that
U(a) = U(b) = E ;
U(x) < E ,
x Î (a, b) ;
U ¢( a ) = 0 ;
U ¢¢( a ) < 0 ;
U ¢( b ) > 0 .
··
Then system (7.1) corresponding to x + U ¢( x ) = 0 possesses an orbit homoclinic to ( a , 0 ) that passes through the point ( b , 0 ) .
Unfortunately, as the cycle of Exercises 7.3–7.5 shows, the homoclinic orbit of autonomous equation (7.1) cannot be used directly to construct a discrete dynamical
system with a transversal homoclinic trajectory.
E x e r c i s e 7.3 For every t > 0 define the mapping f : R n ® R n by the formula f ( x 0 ) = x ( t ; x0 ) , where x ( t ; x0 ) is a solution to equation
(7.1) with the initial condition x0 . Show that f is a diffeomorphism
in R n with a fixed point z 0 and a family of homoclinic orbits
{ yns : n Î Z } , where yn = z ( s + n t ) , s Î [ 0 , t) .
E x e r c i s e 7.4 Prove that the derivative f ¢ of the mapping f constructed in
Exercise 7.3 can be evaluated using the formula f ¢( x0 ) w = y ( t , w) ,
where y ( t , w ) is a solution to problem
y· ( t ) = g ¢ ( x ( t ) ) y ( t ) ,
y ( 0 ) = w0 .
Here x ( t ) = x ( t ; x0 ) is a solution to equation (7.1) with the initial
condition x0 .
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Homoclinic Chaos in Infinite-Dimensional Systems
Let f be the mapping constructed in Exercise 7.3 and let
{ z ( t ) : t Î Z } be a homoclinic orbit of equation (7.1). Show that
{ wn = z· ( n t ) : n Î Z } is a bounded solution to the difference equation wn + 1 = f ¢ ( yn ) wn , where yn = z ( n t ) (Hint: the function
w ( t ) = z· ( t ) satisfies the equation w· = g ¢( z ( t ) ) w ).
E x e r c i s e 7.5
Thus, due to Theorems 2.1 and 3.1 the result of Exercise 7.5 implies that the set
L = { z0 } È { z ( n t ) : n Î Z }
cannot be hyperbolic with respect to the mapping f defined by the formula
f ( x 0 ) = x ( t ; x0 ) , where x ( t ; x0 ) is a solution to equation (7.1) with the initial
condition x0 . Nevertheless we can indicate some quite simple conditions on the
class of perturbations { h ( t , x , m ) } periodic with respect to t under which the monodromy operator of the problem
x ( t ) Î Rn ,
(7.2)
x· ( t ) = g ( x ( t ) ) + m h ( t , x ( t ) , m ) ,
possesses a transversal (hyperbolic) homoclinic trajectory for m small enough.
Further we will use the notion of exponential dichotomy for ordinary differential
equations (see [15], [16] as well as [5] and the references therein)
Let A( t ) be a continuous and bounded n ´ n matrix function on the real axis.
We consider the problem
=x ,
x· ( t ) = A ( t ) x ( t ) ,
t ÎR, x
(7.3)
t=s
0
in the space X =
It is easy to see that it is solvable for every initial condition.
Therefore, we can define the evolutionary operator F ( t , s ) , t , s Î R , by the formula
Rn .
F ( t , s ) x0 = x ( t ) º x ( t , s ; x0 ) ,
t, s Î R ,
where x ( t ) is a solution to problem (7.3).
E x e r c i s e 7.6
Prove that
F(t , s) = F (t , t) F(t , s) ,
F ( t , t) = I
for all t , s , t Î R and the following matrix equations hold:
d- F ( t , s ) = A ( t ) F ( t , s ) ,
---dt
E x e r c i s e 7.7
d F ( t , s ) = -F ( t , s ) A ( s ) . (7.4)
----ds
Prove the inequality
ìt
ü
ï
ï
F ( t , s ) £ exp í A( t ) dt ý ,
ïs
ï
î
þ
ò
t ³ s.
On the Existence of Transversal Homoclinic Trajectories
Let I be some interval of the real axis. We say that equation (7.3) admits an exponential dichotomy over the interval I if there exist constants K , a > 0 and
a family of projectors { P ( t ) : t Î I } continuously depending on t and such that
P(t) F (t , s) = F (t , s) P (s ) ,
t ³ s;
(7.5)
£ K e -a ( t - s ) ,
t ³ s,
(7.6)
F(t, s) P(s)
F ( t , s ) ( I - P ( s ))
£ K e -a ( s - t ) ,
t £ s,
(7.7)
for t , s Î I .
E x e r c i s e 7.8 Let A( t ) º A be a constant matrix. Prove that equation (7.3)
admits an exponential dichotomy over R if and only if the eigenvalues on A do not lie on the imaginary axis.
The assertion contained in Exercise 7.8 as well as the following theorem on the
roughness enables us to construct examples of equations possessing an exponential
dichotomy.
Theorem 7.1.
Assume that problem (7.3) possesses an exponential dichotomy over
an interval I . Then there exists e > 0 such that equation
x· ( t ) = ( A ( t ) + B ( t ) ) x ( t )
(7.8)
possesses an exponential dichotomy over I , provided B ( t ) £ e for t Î I .
Moreover, the dimensions of the corresponding projectors for (7.3) and
(7.8) are the same.
The proof of this theorem can be found in [15] or [16], for example.
The exercises given below contain some simple facts on systems possessing an exponential dichotomy. We will use them in our further considerations.
E x e r c i s e 7.9
Prove that equations (7.5)–(7.7) imply the estimates
F(t, s) x
³ K -1 e a ( t - s ) ( 1 - P ( s ) ) x ,
F ( t , s ) x ³ K -1 e a ( s - t ) P ( s ) x ,
t ³ s,
t £ s,
for any x Î X º R n .
E x e r c i s e 7.10 Assume that equation (7.3) admits an exponential dichotomy
over R+ = [ 0 , + ¥) . Prove that P ( 0 ) X = V+ , where
ì
ü
V+ º í x Î X º R n : sup F ( t , 0 ) x < ¥ ý
t >0
î
þ
1
a
t
( 1 - P ( 0 ) ) x , t ³ 0 ).
(Hint: F ( t , 0 ) x ³ K e
(7.9)
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Homoclinic Chaos in Infinite-Dimensional Systems
E x e r c i s e 7.11 If equation (7.3) possesses an exponential dichotomy over
R– = ( -¥ , 0 ] , then ( I - P ( 0 ) ) X = V– , where
ì
ü
V– = í x Î X º R n : sup F ( t , 0 ) x < ¥ ý
0
t
<
î
þ
(Hint: F ( t , 0 ) x
³ K -1 e a t P ( 0 ) x ,
(7.10)
t £ 0 ).
E x e r c i s e 7.12 Assume that equation (7.3) possesses an exponential dichotomy over R + (over R – , respectively). Show that any solution to
problem (7.3) bounded on R+ (on R – , respectively) decreases
at exponential velocity as t ® + ¥ (as t ® -¥ , respectively).
E x e r c i s e 7.13 Assume that equation (7.3) possesses an exponential dichotomy over the half-interval [ a , + ¥ ) , where a is a real number.
Prove that equation (7.3) possesses an exponential dichotomy over
any semiaxis of the form [ b , + ¥ ) .
(Hint: P ( t ) = F ( t , a ) P ( a ) F ( a , t ) ).
E x e r c i s e 7.14 Prove the analogue of the assertion of Exercise 7.13 for the
semiaxis ( -¥ , a ] .
E x e r c i s e 7.15 Prove that for problem (7.3) to possess an exponential dichotomy over R it is necessary and sufficient that equation (7.3) possesses an exponential dichotomy both over R+ and R– and has no
nontrivial solutions bounded on the whole axis R .
E x e r c i s e 7.16 Prove that the spaces V+ and V– (see (7.9) and (7.10)) possess the properties
V+ Ç V– = { 0 } ,
V+ + V– = X º R n ,
provided problem (7.3) possesses an exponential dichotomy over R .
E x e r c i s e 7.17 Consider the following equation adjoint to (7.3):
y· ( t ) = -A* ( t ) y ( t ) ,
(7.11)
A* ( t )
where
is the transposed matrix. Prove that the evolutionary
operator Y( t , s ) of problem (7.11) has the form Y( t , s ) = [ F( s , t ) ] * .
E x e r c i s e 7.18 Assume that problem (7.3) possesses an exponential dichotomy over an interval I . Then equation (7.11) possesses exponential
dichotomy over I with the same constants K , a > 0 and projectors
Q (t) = I - P(t)* .
E x e r c i s e 7.19 Assume that problem (7.3) possesses an exponential dichotomy both over R + and R – . Let dim V+ + dim V– = n , where V± are
defined by equalities (7.9) and (7.10). Show that the dimensions of
the spaces of solutions to problems (7.3) and (7.11) bounded on the
whole axis are finite and coincide.
On the Existence of Transversal Homoclinic Trajectories
E x e r c i s e 7.20 Assume that problem (7.3) possesses an exponential dichotomy over R . Then for any s Î R and t > 0 the difference equation
xn = F ( s + t n , s ) xn - 1 possesses an exponential dichotomy over
Z (for the definition see Section 2).
Let us now return to problem (7.1). Assume that z0 is a hyperbolic fixed point for
(7.1), i.e. the matrix g ¢( z0 ) does not have any eigenvalues on the imaginary axis. Let
z ( t ) be a trajectory homoclinic to z0 . Using Theorem 7.1 on the roughness and the
results of Exercises 7.13 and 7.14 we can prove that the equation
y· = g ¢ ( z ( t ) ) y
(7.12)
possesses an exponential dichotomy over both semiaxes R + and R – . Moreover, the
dimensions of the corresponding projectors are the same and coincide with the dimension of the spectral subspace of the matrix g ¢ ( z 0 ) corresponding to the spectrum in the left semiplane. Therewith it is easy to prove that dim V+ + dim V– = n ,
where V± have form (7.9) and (7.10). The result of Exercise 7.15 implies that equation (7.12) cannot possess an exponential dichotomy over R ( y ( t ) = z· ( t ) is a solution to (7.12) bounded on R ) while Exercise 7.19 gives that the number of linearly
independent bounded (on R ) solutions to (7.12) and to the adjoint equation
y· = -[ g ¢ ( z ( t ) ) ] * y
(7.13)
is the same. These facts enable us to formulate Palmer’s theorem (see [14]) as follows.
Theorem 7.2.
Rn
Assume that g ( x ) is a twice continuously differentiable function from
into R n and equation
x· = g ( x )
possesses a fixed hyperbolic point z0 and a trajectory { z ( t ) : t Î R } homoclinic to z 0 . We also assume that y ( t ) = z· ( t ) is a unique (up to a scalar factor) solution to equation
y· = g ¢ ( z ( t ) ) y
(7.14)
bounded on R . Let h ( t , x , m ) be a continuously differentiable vector function T -periodic with respect to t and defined for t Î R , x - z ( t ) < D0 ,
m < s 0 , m Î R . If
¥
ò (y(t) , h(t, z(t) , 0) )
-¥
¥
Rn
dt = 0 ,
ò (y(t) , h (t, z(t) , 0) )
t
Rn
dt ¹ 0 ,
(7.15)
-¥
where y ( t ) is a bounded (unique up to a constant factor) solution to
the equation adjoint to (7.14),, then there exist D and s such that for
0 < m < s the perturbed equation
x· = g ( x ) + m h ( t , x , m )
(7.16)
407
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Homoclinic Chaos in Infinite-Dimensional Systems
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possesses the following properties:
(a) there exists a unique T -periodic solution x 0 ( t , m ) such that
z0 - x0 ( t , m ) < D ,
t ÎR ,
and
6
sup z 0 - x 0 ( t , m ) ® 0 ,
t ÎR
m®0 ;
(b) there exists a solution x ( t , m ) bounded on R and such that
x (t, m ) - z(t) < D ,
t ÎR ,
sup x ( t , m ) - z ( t ) = 0 ( m ) ,
t ÎR
and
lim
t ® ±¥
x ( t , m ) - x0 ( t , m ) = 0 ;
(c) the linearized equation
ì
ü
y· = í g ¢ ( h ( t , m ) ) + m h ¢x ( t , h ( t , m ) , m ) ý y ,
(7.17)
î
þ
where h ( t , m ) is equal to either x ( t , m ) or x 0 ( t , m ) , possesses an exponential dichotomy over R .
This theorem immediately implies (see Exercise 7.20 and Theorem 3.1) that
under conditions (7.15) the monodromy operator for problem (7.16) has a hyperbolic fixed point in a vicinity of the orbit { z ( t ) : t Î R } and a transversal trajectory homoclinic to it.
We will not prove Theorem 7.2 here. Its proof can be found in paper [14]. We only
outline the scheme of reasoning which enables us to construct a homoclinic trajectory x ( t , w ) . Here we pay the main attention to the role of conditions (7.15). If we
change the variable x = z ( t ) + z in equation (7.16), then we obtain the equation
·
z = g(z(t) + z) - g(z(t) ) + m h(t , z(t) + z , m) .
We use this equation to construct a mapping F from Cb1 ( R , R n ) ´ R into
C b0 ( R , R n ) acting according to the formula
·
( z , m ) ® F ( z , m ) = z - { g ( z ( t ) + z ) - g ( z ( t ) ) + m h ( t , z ( t ) + z , m ) } . (7.18)
We remind that Cbk ( R , X ) is the space of k times continuously differentiable bounded functions from R into X with bounded derivatives with respect to t up to the k th order, inclusive.
Thus, the existence of bounded solutions to problem (7.16) is equivalent to the
solvability of the equation F ( z , m ) = 0 . It is clear that F ( 0 , 0 ) = 0 . Therefore, in
order to construct solutions to equation F ( z , m ) = 0 we should apply an appropriate version of the theorem on implicit functions. Its standard statement requires
that the operator L = Dz F ( 0 , 0 ) be invertible. However, it is easy to check that the
operator L has the form
On the Existence of Transversal Homoclinic Trajectories
( L y ) ( t ) = y· ( t ) - g ¢( z ( t ) ) y ( t ) .
Therefore, it possesses a nonzero kernel ( L z· = 0 ) . Hence, we should use the modified (nonstandard) theorem on implicit functions (see Theorem 4.1 in [14]). Roughly
speaking, we should make one more change z = m w and consider the equation
F (m w, m) = 0 .
(7.19)
If this equation is solvable and the solution w depends on m smoothly, then w satisfies the equation
D z F ( m w , m ) [ w + m wm ] + Dm F ( m w , m ) = 0 ,
(7.20)
where wm is the derivative of w with respect to the parameter m . This equation can
be obtained by differentiation of the identity F ( m w ( m ) , m ) = 0 with respect to m .
Due to the smoothness properties of the mapping F , it follows from (7.20) the solvability of the problem
Dz F ( 0 , 0 ) w 0 + Dm F ( 0 , 0 ) = 0 ,
which is equivalent to the differential equation
w· 0 = g ¢ ( z ( t ) ) w0 + h ( t , z ( t ) , 0 )
(7.21)
(7.22)
in the class of bounded solutions. It is easy to prove that the first condition in (7.15)
is necessary for the solvability of (7.22) (it is also sufficient, as it is shown in [14]).
Further, the necessary condition of the dichotomicity of (7.17) for h ( t , m ) =
= x ( t , m ) on R is the condition of the absence of nonzero solutions to equation
(7.17) bounded on R . However, this equation can be rewritten in the form
Dz F ( m w , m ) y ( m ) = 0 ,
(7.23)
where w = w ( m ) is determined with (7.19). If we assume that equation (7.23) has
nonzero solutions, then we differentiate equation (7.23) with respect to m at zero,
as above, to obtain that
Dz z F ( 0 , 0 ) w0 + Dz m F ( 0 , 0 ) y0 + Dz F ( 0 , 0 ) y1 = 0 ,
(7.24)
where w0 is a solution to equation (7.21), y0 = y ( 0 ) , y1 = Dm y ( 0 ) , and y ( m ) is
a solution to equation (7.23). Equation (7.17) transforms into (7.14) when m = 0 .
Therefore, the condition of uniqueness of bounded solutions to (7.14) gives us that
y0 = c0 z· ( t ) , therewith we can assume that c 0 = 1 . Hence, equation (7.24) transforms into an equation for y1 of the form
y· - g ¢( z ( t ) ) y = a ( t ) ,
(7.25)
1
1
where
a ( t ) = -æ Dzz F ( 0 , 0 ) w0 + Dz m F ( 0 , 0 ) z· ö ( t ) =
è
ø
= Dx x g ( z ( t ) ) w0 ( t ) + Dx h ( t , z ( t ) , 0 ) z· ( t ) .
409
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Homoclinic Chaos in Infinite-Dimensional Systems
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Here w0 ( t ) is a solution to (7.22). A simple calculation shows that equation (7.25)
can be rewritten in the form
6
d
----- ( y1 ( t ) + w· 0 ( t ) ) - g ¢( z ( t ) ) ( y1( t ) + w· 0 ( t ) ) = ht ( t , z ( t ) , 0 ) .
dt
(7.26)
The second condition in (7.15) means that equation (7.26) cannot have solutions
bounded on the whole axis. It follows that equation (7.23) has no nonzero solutions,
i.e. equation (7.17) is dichotomous for h ( t , m ) = x ( t , m ) .
Thus, the first condition in (7.15) guarantees the existence of a homoclinic trajectory x ( t , m ) while the second one guarantees the exponential dichotomicity of
the linearization of the equation along this trajectory.
As to the existence and properties of the periodic solution x 0 ( t , m ) , this situation is much easier since the point z 0 is hyperbolic. The standard theorem on implicit functions works here.
It should be noted that condition (7.15) can be modified a little. If we consider a
“shifted” homoclinic trajectory zs ( t ) = z ( t - s ) for s Î R instead of z ( t ) in Theorem 7.2, then the first condition in (7.15) can be rewritten in the form
¥
D(s) º
ò æèy(t - s) , h(t, z(t - s) , 0)öø
Rn
dt = 0 .
-¥
If we change the variable t ® t + s , then we obtain that
¥
D(s) =
ò æèy(t) , h(t + s, z(t) , 0)öø
Rn
dt .
(7.27)
-¥
It is evident that
¥
D¢( s ) =
ò æèy(t - s) , h (t, z(t - s) , 0)öø
t
Rn
dt .
-¥
Therefore, the second condition in (7.15) leads us to the requirement D¢( s ) ¹ 0 .
Thus, if the function D ( s ) has a simple root s0 ( D ( s 0 ) = 0 , D¢( s0 ) ¹ 0 ), then the
assertions of Theorem 7.2 hold if we substitute the value z ( t - s0 ) for z ( t ) in (b).
Performing the corresponding shift in the function x ( t , m ) , we obtain the assertions of the theorem in the original form. Thus, condition (7.15) is equivalent to the
requirement
D ( s0 ) = 0 ,
D ¢( s 0 ) ¹ 0
for some s0 Î R ,
(7.28)
where D ( s ) has form (7.17).
In conclusion we apply Theorem 7.2 to prove Theorem 6.1. The unperturbed Duffing
equation can be rewritten in the form
·
ì x1 = x2 ,
(7.29)
í ·
î x2 = b x1 - a x13 .
On the Existence of Transversal Homoclinic Trajectories
The equation linearized along the
cise 7.1) has the form
·
ì y1 =
í ·
îy =
2
homoclinic orbit z ( t ) = (h ( t ) , h· ( t ) ) (see Exer-
y2 ,
b y1 - 3 ah ( t ) 2 y1 .
(7.30)
Let us show that system (7.30) has no solutions which are bounded on the axis and
not proportional to z· ( t ) . Indeed, if w ( t ) = ( v ( t ) , v· ( t ) ) is another bounded solution,
·· ( t ) ® 0 as t ® ¥ , the Wronskian W ( t ) =
then due to the fact that h· ( t ) + h
·
··
·
= v ( t ) h ( t ) - v ( t ) h ( t ) possesses the properties
d
----- W ( t ) = 0
dt
and
lim W ( t ) = 0 .
t ®¥
This implies that W ( t ) º 0 and therefore v ( t ) is proportional to h· ( t ) .
Evidently, the equation adjoint to (7.30) has the form
·
2
ì y1 = - b y2 + 3 a h ( t ) y2 ,
í ·
î y 2 = -y1 .
(7.31)
Since we have that
y·· 2 - b y2 + 3 a h ( t )2 y2 = 0 ,
·· ( t) ; h· ( t) ) . Let us now
a solution to (7.31) bounded on R has the form y ( t ) = ( -h
consider the corresponding function D( s ) . Since in this case h ( t , x , m ) =
= (0 ; f cos w t - d x2 ) , we have
¥
D(s) =
ò h· (t) [ f cos w (t + s) - dh· (t)] dt ,
-¥
where
h(t) =
2b
------- sech b t =
a
-1
8b
------- æ e b t + e - b tö .
a è
ø
Calculations (try to do them yourself) give us that
sin w s
2- × ------------------------------4- d b 3/2 .
- - -----D ( s ) = 2 f w --a
3
a
p
w
cosh æ ------------ ö
è2 bø
Therefore, equation D ( s ) = 0 has simple roots under condition (6.19). Thus, the assertion of Theorem 6.1 follows from Theorem 7.2.
It should be noted that in this case the function D ( s ) coincides with the famous
Melnikov function arising in the geometric approach to the study of the transversality (see, e.g., [1], [2] and the references therein). Therewith conditions (7.28) transform into the standard requirements on the Melnikov function which guarantee the
appearance of homoclinic chaos.
411
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Homoclinic Chaos in Infinite-Dimensional Systems
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Addition to the English translation:
6
The monographs by Piljugin [1*] and by Palmer [2*] have appeared after publication
of the Russian version of the book. Both monographs contain an extensive bibliography and are closely related to the subject of Chapter 6.
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systems // Siberian Math. J. –1988. –Vol. 29. –#3. –P. 408–417.
11. HALE J. K., LIN X.-B. Symbolic dynamics and nonlinear semiflows // Ann. Mat.
Pura ed Appl. –1986. –Vol. 144. –P. 229–259.
12. LANI-WAYDA B. Hyperbolic sets, shadowing and persistance for noninvertible
mappings in Banach spaces. –Essex: Longman, 1995.
13. HOLMES P., MARSDEN J. A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam // Arch. Rat. Mech Anal.
–1981. –Vol. 76. –P. 135–165.
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J. Diff. Eqs. –1984. –Vol. 55. –P. 225–256.
15. HARTMAN P. Ordinary differential equations. –New York; London; Sydney: John
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1*. PILJUGIN S. JU. Shadowing in dynamical systems. Lect. Notes in Math. 1706.
–Berlin; Heidelberg; New York: Springer, 1999.
2*. PALMER K. Shadowing in dynamical systems. Theory and Applications.
–Dordrecht; Boston; London: Kluwer, 2000.